Welcome to Webmath Personal!

Introduction

This version of Webmath, adapted from the popular math-help web site at http://www.webmath.com, preserves our design philosophy of software that offers math help on a computer:

give you help with the particular problem you’re working on.

Whether it be on our web site or with this software, you won’t find any “drill” questions with predetermined answers, or “practice tests” here.  Our math help software focuses on finding the solution to your math problem.  In fact, unless you present this software with some math problem, it will be of little use!  It works only when you present it with a particular math problem to solve, like one from your very own math book.  Once you do so, this software will instantly analyze your problem and give you step-by-step solution to it!

This software offers a wide range of problem solving capabilities, and is best suited for a student studying Algebra One or Algebra Two.  More math problem solving capabilities (at levels both above and below algebra) can be found for instant, online use at our website http://www.webmath.com.

To continue on with this online help, here are two links you can go on with.

See the Table of Contents

Tips on typing your problem into this software


Tips on typing your problem into this software

As you use Webmath Personal, you will eventually be presented with a window containing type-in boxes, into which you are to type your math problem for the system to solve.

While reading through this online help, especially when you click a “What can I type here?” button, many specific example problems are given.  These examples are meant to be directly typed into the window for which you requested help.  Any information that you are meant to type into a window box is shown in a red color.

Here are some tips for entering your own problem:

•	Most numbers and variables (letters) you have in a math problem can just be entered as is (like a,b,c, A, B, C, 1, 2, 5, 66.2, etc.)

•	The + sign on your keyboard means “plus,” and the - sign on your keyboard means minus.

•	The / (forward slash) on your keyboard means “divided by.”  So, 3 divided by 2 would be entered 3/2.  This slash also serves as a fraction bar, any time you enter a fraction.  So

 is typed as x/2.  Likewise, [image] is typed like (3x+2z)/(16a).  Note here how the parentheses help to group the numerator and denominator around the fraction bar (the “/”).

•	The * symbol on your keyboard (shift the 8-key) means multiply.  So 5 times 2 would be entered as 5*2.  However, most of the time you don’t have to enter anything for multiplication, because the software can figure out on its own when two things are supposed to be multiplied.  For instance, if you need to multiply 3 times x, you can enter it like 3*x, but entering it more naturally like 3x is O.K., because the system can figure out that this means “3 times x.”  The same goes for 3*(x-2), which can be entered like 3(x-2).  Obviously, you should put the * symbol between pure numbers, because for example, 52 is not the same at 5*2 (=10).

•	If you need to type an exponent, as in x2, you do so using the caret, or ^ symbol (shift the 6-key on most keyboards).  So, x2 would be entered like x^2.  In a more complicated example, 3x2-5x+6 would be entered like this: 3x^2-5x+6.  Or, (x+6)3 would be entered like this: (x+6)^3.

•	If you want to use any trigonometric functions, most standard ones, are recognized, like these:

•	cos(x) for the cosine,

•	sin(x) for the sine,

•	tan(x) for the tangent,

•	asin(x) for the arcsine,

•	acos(x) for the arccosine,

•	atan(x) for the arctangent,

•	sinh(x) for hyperbolic sin,

•	cosh(x) for hyberbolic cosine, and

•	tanh(x) for the hyperbolic tangent.

For the trigonometric functions, all angles must be input into them in radians.  If you need to do it in degrees, don’t despair!  Radian and degrees are related like this: radians=3.14*degrees/180.  So, for instance, if you needed the cosine of 60 degrees, you would type cos(60*3.14/180).  Likewise, the sin of 10 degress would be sin(10*3.14/180).  That’s it!

Other recognized functions include:

•	exp(x) for ex,

•	log(x) for the ln(x) (natural log),

•	log10(x) for log with a base of 10,

•	sqrt(x) for [image] •	, and

•	abs(x) for the absolute value of x, or


WebMath Personal Help

Table of Contents

Calculate

Without Variables

With Variables

Plotting

XY or Cartensian Plots

Polar Plots

Data Files

Simplify

Powers

Products

Like TermsAnything Else

Polynomials

Adding and Subtracting

Multiply

Divide

Raising to some power

Factoring a Polynomial

Factor by finding the GCF

Factor a difference of two squares

Factor a trinomial

Quadratic Equations

Solve by factoring

Solve using the Quadratic Formula

Radicals

Square roots

Other roots

Equation Solving

One equation and one unknown

Two equations and two unknowns

Word Problems

Solving Word Problems


Simplify Powers

This selection will simplify an expression involving variables, exponents, or any combination of these items.

The purpose of solving exercises like these is to demonstrate that you know how to combine the exponents of terms that have the same base.  Also, that you know not to combine the exponents of terms that do not have the same base.

The general rule applied to problems here is that:

.

(there is nothing to combine here, because the bases are different; you just leave the expression as it)

and

 (since the bases are the same here, you add the exponents)

and

(to raise a term with an exponent to a second exponent, just mulitply the exponents together).

Here are some examples of what you could type in the input box here:

•	Type x^2x^3 to simplify the expression

•	Type x^2y^8 to simplify the expression

•	Type a^2(x^2b^3)^2 to simplify the expression

Tips on typing your problem into this software


Simplify Products

This selection will simplify “products of terms,” or terms that are to be multiplied together.  It will combine exponents, combine coefficients, and remove parentheses or other grouping symbols.

  Here are some examples of what you could type here:

•	Type (a^2b^3)(3a^2) to simplify the expression

•	Type (4x)(3a^2)(16z^2x) to simplify the expression

Tips on typing your problem into this software


Simplify Like Terms

This selection will recognize and simplify “like terms” in an expression.  Like terms are terms that contain the same “literal” (or variable) parts.  In other words, like terms are terms that contain the same variables with the same exponents.

It is important to be able to simplify like terms because many times an algebraic expression is not written in simplest form, and is therefore not as concise and “final” of an expression as it could be.

Like terms can only be combined (added or subtracted) if the variable and exponent parts of the terms are identical.

Here are some examples of what you could type here:

•	Type 2x+4x to simplify the expression [image] •	.

•	Type 3a-5a+a to simplify the expression

•	Type a^2b-3a^2b to simplify the expression

•	Type 5ab+2c-2ba to simplify the expression

Tips on typing your problem into this software


Simplify Anything Else

This selection will not try to recognize anything in particular about the problem you type in.  Instead, every algebraic operation programmed into this software will be applied to your problem, in an attempt simplify it.

Here are some examples of what you could type here:

•	Type -(1-(x+1))  to simplify the expression

•	Type a-(b-c) to simplify the expression

•	Type [8+2(a-b)-x]-y to simplify the expression

Tips on typing your problem into this software


Adding and Subtracting Polynomials

This selection will add or subtract polynomials.

Here are some examples of what you could type here:

•	Type (3x^2+5x-4)+(2x+5)  to add the polynomials

 and

•	Type (-4x^3+8x^2-2x-3)-(4x^3-7x^2+6x+5) to subtract the polynomials

 and

•	Type (2m^2-m+4)+(3m^2+m-5) to add the polynomials

 and

•	Or, just type in your problem!

Tips on typing your problem into this software


Multiplying Polynomials

This selection will multiply polynomials together.  If the problem you present is that of two binomials being multiplied together, the F.O.I.L. method will be applied and demonstrated in the solution.

Here are some examples of what you could type here:

•	Type (x+1)(x-6) to multiply the polynomials

 and

•	Type (2y-7)(7x-6) to multiply the polynomials

 and

•	Type (3x^2+5x-4)(2x+5)  to multiply the polynomials

 and

•	Type (2m^2-m+4)(3m^2+m-5) to add the polynomials

 and

•	Or, just type in your problem!

Tips on typing your problem into this software


Dividing Polynomials

This selection will divide polynomials.  At this time however, this facility is somewhat limited.  It can only divide a polynomial by a single term (a monomial).  Future versions of both this program and the WebMath.com web site will be able to handle more complicated polynomial division (e.g. like synthetic division).  Check back to the web site periodically for updates.

Here are some examples of what you could type here:

•	Type (6x^2+2x)/(2x) to divide the polynomial

by the term

•	Type (8y-4)/2 to divide the polynomial

by the number 2.

•	Type (16x^5-4x^4+8x^3)/(16x^3) to divide the polynomial

 by

•	Or, just type in your problem!

Tips on typing your problem into this software


Raising a Polynomial to Some Power

This selection will show you how to raise a polynomial to some power.

Here are some examples of what you could type here:

•	Type (x+1)^2 to raise the polynomial [image] •	 to the power 2

(Another way to state this: “To square  x+1.”)

•	Type (8y-4)^4 to raise the polynomial [image] •	 by the power 4.

•	Type (2x^2+3x+1)^3 to raise polynomial [image] •	 to the 3rd power.

(Another way to state this: “To cube [image] ”)

•	Or, just type in your problem!

Tips on typing your problem into this software


Factoring a Polynomial by finding its Greatest Common Factor (GCF)

This selection will try to factor a polynomial by finding its Greatest Common Factor (GCF).  The GCF of a polynomial is the largest term that is a factor of each term in the polynomial.  Or, in other words, the GCF of a polynomial is the largest term that evenly divides into each term of the polynomial.

The final factored form of a polynomial factored in this way will be the GCF times a new polynomial, which is the orginial polynomail with the GCF factored out.

Here are some examples of what you could type here:

•	Type 15x^3+9x to factor the polynomial

by finding its GCF.

•	Type 12x^3-16x^5 to factor the polynomial

•	Type 2x^4+4x^3-8x^2 to factor the polynomial

•	Or, just type in your problem!

Tips on typing your problem into this software


Factoring a Polynomial that is the difference of two squares

This selection will try to factor a polynomial that is the difference of two perfect squares.  A polynomial entered into this selection must have the following two properties:

1. 	It must contain only two terms, and

2. 	each term must be a perfect square.  That is, each term must be

the result of multiplying another term by itself..

If both of these conditions are not met, then a polynomials cannot be factored as the “difference of two squares.”

Here are some examples of what you could type here:

•	Type m^2-n^2 to factor the polynomial

.

Note that this polynomial contains two terms, and each term is a perfect square, because [image], and

•	Type x^2-9 to factor the polynomial

.

Note that this polynomial contains two terms, and each term is a perfect square, because [image], and [image].

•	Type 144w^4x^6-121y^8z^2 to factor the polynomial

.

Note that this polynomial also contains two terms, and each term is a perfect square, because

,

and

.

•	Or, just type in your problem!

Tips on typing your problem into this software


Factoring a Trinomial (a polynomial that has three terms)

This selection will try to factor a trinomial, or a polynomial that has three terms.  Many trinomials simply are not factorable, but still many others can be factored into the product of two binonials (polynomials with two terms).

Any polynomial entered into this selection must have three terms.

Here are some examples of what you could type here:

•	Type x^2+7x+10 to factor the polynomial

.

Note that this polynomial has three terms, and can be factored into the product of two binomials.  This product will look like this: (x+??)(x+??).

(Hint: Type the problem in to find out what the ?? symbols should be!)

•	Type x^2+5x+6 to factor the polynomial

.

Note that this polynomial contains three terms.  Try it out!  It can be factored into the product of two binomial.

•	Type b^2-11bd-60d^2 to factor the polynomial

.

•	Or, just type in your problem!

Tips on typing your problem into this software


Solving a Quadratic Equation by Factoring it

This selection will try to find the solutions to a quadratic equation by factoring it.  This method of solving a quadratic equation may not always work, since not all quadratic equations are factorable in the first place.

The form of the quadratic equation you enter here must be in the form:

“Math-expression” = 0,

where “Math-expression” contains all of the terms of the quadratic function you wish to solve.  The reason why these equations are called “quadratic equations,” is for the following reasons:

1.  They are “quadratic” because they contain a variable (like a,b,c,x,or y, etc.) raised to the power 2, like [image] or [image] etc.

2.  They are “equations” because they have the form of “Math-expression” = 0.  The “=0” makes them an equation.

Here are some examples of what you could type here:

•	Type x^2+x-6 to solve the quadratic equation

.

•	Type x^2+5x+6 to solve the quadratic equation

.

•	Type 2x^2-x-1 to solve the quadratic equation

.

•	Or, just type in your quadratic equation!

Tips on typing your problem into this software


Solving a Quadratic Equation using the Quadratic Formula

For those quadratic equations that can’t be solved by factoring, there is a chance that they can be solved using the Quadratic Formula.  Here’s how to use the quadratic formula:

Given a quadratic equation in this form:

,

where x is the variable of the equation (it could just as easily be any other letter of the alphabet as well), A is the coefficient of the squared term, B is the coefficient of the linear term (variable to the first power), and C is the constant term.  The solutions, or roots to this equation that will make the “left hand side” of the equation equal to zero can be found from the following formula:

The [image] signs mean that the actual two solutions can be found from the following two equations:

Solution #1 from

Solution #2 from

In an actual problem all you have to do is figure out what A, B, and C are, plug them into the above formulas, and you can calculate the two solutions to the quadratic equation.  It is this approach to solving quadratic equations that this selection uses.

In order to use this selection, you must arrange the form of your quadratic equation to be like this:

“Math-expression” = 0,

where “Math-expression” contains all of the terms of your quadratic function you wish to solve.

Here are some examples of what you could type here (note that these quadratic equations cannot be solved by factoring):

•	Type A=3, B=-1, C=-2 to solve the quadratic equation

•	Type A=1, B=-4, C=1 to solve the quadratic equation

•	Or, just type in your quadratic equation!

Tips on typing your problem into this software


Calculating with no variables

Most of us are used to doing arithmetic with numbers, that is multiplying, dividing, adding, subtracting, and raising number to different powers.  This menu option basically serves as a numerical calculator for performing just such “numerical work.”  Only, instead of the buttons ususally seen on a calculator, you can enter your mathematical expression in a full “algebra-like” format.  Once done, you expression will be evaluated step by step until the final, single number result is found.  This process is called the “evaluation of mathematical expressions.”

What is important to know here is the “precedence of operators.”  That is, if you have an algebraic expression involving a large combination of +, -, times, and divide, in what order are the signs supposed to be evaluated?  There is a logical and correct order to follow, called the “order of operations.”  It goes like this:

1.	First, powers and roots are done (that is, do the exponents first).

2.	Second, multiplication and division are done in order, from left to right.

3.	Third, addition and subtraction are done, in order, from left to right.

Grouping symbols, like ( and ) change the order or operation.  In fact, you must always do what is inside of grouping symbols first.  Once working inside grouping symbols, just treat the inside expression like a little “mini” problem in itself, in which steps 1 - 3 above are always followed.

Here are some examples of what you could type here:

•	Type: 2+2

•	Type (2+8)/4

•	Type 13-9+16-8

•	Type:4*3+15/5 to simplify

•	Type: (-5)^2+(-2)(6)-(-4)^2 to simplify

•	Or, just try your own expression!

Tips on typing your problem into this software


Calculating with variables

Doing calculations with variables is just like doing calculations without variables, only you have to get rid of the variables first.

What is a variable?  In algebra, variables are used to represent numbers.  It is usually a letter from A to Z and is sort of like a “placeholder” for a number.  Before actually doing a numerical calculation with variables, the variables must be replaced with what numbers they are supposed to stand for in an expression.

To evaluate an expression having variables and numbers:

1.	Replace each variable by its number value, usually enclosing the number in parentheses.

2.	Carry out all arithmetic operations, using the correct order of operations.

Essentially, once you complete step 1, and have all of the variables replaced with numbers, step two is the entire process that is described on doing calculations without variables.  With all of the variables “gone” (replaced with their associated numbers), you are left with a normal “numbers only” calculation to perform.

This menu option allows you to enter a mathematical expression containing variables.  In all of the little boxes below the main entry box, you are to fill in what variables are to stand for what numbers. You don’t need to fill in all of the boxes, just the ones of the variables you use in your expression.  When you click “Calculate It!”, this selection will first replace all variables in your expression with their proper values, then proceed to evaluate the expression, to arrive at the final, numerical answer.

Here are some examples of what you could type here:

•	Type: 2+a, with a 2 put in the little box next to the letter a.

•	Type (x+y)/4, with a 1 put in the little box next to the x and a 1 put in the little box next to the y.

•	Type 3b, with a -5 put in the little box next to the b.

•	Type:4+a(x+y), with a 3 put in the little box next to the a, a 4 put in the box next to the x, and a -7 put in the box next to the y.

Or, just try your own expression!

Tips on typing your problem into this software


Radicals with square roots

The square root of a number p is a number, that, when squared (multiplied by itself) gives p.  For example, 3 is the square root of 9, because if you multiply 3 by itself (3 times 3), you get 9.  The mathematical symbol for a square root is a sign that looks like this:.  This sign is sometimes called a “radical” sign, hence the name of this section.

The math specialty of this section is in simplifying expressions under (or inside) a radical sign.  For instance, if you see an expression like [image], it basically is asking the question “what is the square root of 9?”  In this case, it is not written in the simplest, compact form, because, as we mentioned above 3*3 is 9, so [image] =3.

Likewise, since Algebra very often can involve variables, so you might have an expression like this: .  This is also not written in the simpilest possible form, because the square root of “x squared” (which is what this is asking) is just x.  So instead of writing [image], we could just write x.  That would be the simplest way to write this expression.

This selection will help you to simplify expressions under a radical sign.  Here are some examples of what you can type here (note how the radical sign is already provided for you in the window; you just have to type in what is under the radical sign in your problem):

•	You type just a 9 to work on the problem [image] •	, see if the square root of 9 can be simplifed.

•	You type just 17 to see if [image] •	 can be simplified (the square root of 17).

•	Type type just 25ab^2 to see if [image] •	can be simplified

•	Type 250x^5y^2z to see if [image] •	 can be simplified

•	Type x^3 to see if [image] •	can be simplified.

•	Or just try your own problem!

Tips on typing your problem into this software


Radicals without square roots

This selection is a lot like the selection on simplifying  square roots.  They differ in that this section can handle any root, not just “square roots” which involve the power of 2 (like 3*3 is 9, or [image] is [image]).  This selection will help you to simplify expression involving cube roots (to the 3rd power) and higher.

In this selection, you need to type two items into the window.  The first is the expression you wish to simplify, which goes under the radical sign (the [image]).  The other is the root itself, which is some number, like 1, 2, 3, 4, etc. that is the level of the root itself to figure out.  This goes in the little box towards the upper left corner of the window.  For instance, if you entered a 3 in the little box, and a 27 under the radical sign, you would be requesting the problem [image] to be solved, or “what is the cube root of 27?”

This selection will help you to simplify expressions like these under a radical sign, with some “root” value other than 2, as is fixed in the square rootscaseHere are some examples of what you can type here.  Note how the radical sign is already provided for you in the window; you just have to type in what is under the radical sign in your problem, and what root to take in the little box in the upper left:

•	You type just a 27 under the radical sign, and a 3 in the little box towards the upper left work on the problem [image] •	and see if the cube (3rd) root of 27 can be be simplifed.

•	You type just x^3 under the radical sign and a 3 in the little box towards the upper left to see if [image] •	can be simplified.

•	You type just 4x^3y^8, and a 3 in the little box to see if [image] •	 can be simplified (the cube root of [image] •	).

•	Or just try your own problem!

Tips on typing your problem into this software


Solving an equation with one unknown

Being able to solve equations is a very important part of learning algebra.  This is because mathematics is a problem solving tool, and most problems can be formulated (or at least thought of ) in terms of equations.

This selection will show you how to solve a single equation that contains a single unknown variable.  An equation is usually made up of three parts.

•	An equal sign.  This is the sign that looks like this: =.  This is the mathematical symbol that means two things are equal to each other.  The two things that are said to be equal to each other are what appears on either side of the equal sign, that is, on the left hand side, and on the right hand side.

•	The left hand side.  Any algebraic expression all contained on the left hand side of the equal sign.

•	The right hand side.  Any algebraic expression all contained on the right hand side of the equation.

The left and right hand sides of an equation are sort of “separated” by the equal sign, but individual parts of either side may be moved freely from one side to the other following the usual rules of algebra.  In particular, since the equal sign between the two sides means one side is “the same” (equal to) the other side, then what you do mathematically to one side of the equation, you must also do to the other side of the equation, in order to preserve the equality.  For example, say you wanted to add a 3 to the left side of an equation.  You are perfectly free to do so, as long as you also add a 3 to the other side of the equation as well.

The main lesson to be learned in solving an equation is to be able to properly, according to the rules of algebra, arrange the left and right hand sides so that the left hand side eventually contains only the unknown variable, and the right hand side contains all of the other known items, usually a bunch of numbers and mathematical operators (like +, -, *, /, etc.)

This selection assumes you have one equation to solve.  Your equation should have a left hand side and right hand side, with an equal sign in between.  There should only be one unknown you are interested in solving for in your equation.

In the window, you should be able to see the equal sign.  Type the left hand side of your equation into the box to the left of the equal sign, and the right hand side of your equation into the box at the right of the equal sign.  Also, type the unknown variable to solve for into the small box below the main equation line.  Once done, just click “Solve it!”

Here are some examples of what you could type here:

3x     =     12

What variable do you want to solve for? x

y+6     =     -7y

What variable do you want to solve for? y

7a+b     =     2

What variable do you want to solve for? a

Note that this equation has two variables in it, a and b.  We have only asked it to solve for variable a, which it will do, but the answer will contain the b as well (the b just doesn’t disappear).

4x+14+2x     =     12-3x-8

What variable do you want to solve for? x

10-7z     =     4(11-6z)

What variable do you want to solve for? z

(10-7z)/4     =     11z+6

What variable do you want to solve for? z

This example will solve the equation

.  Notice how the parentheses ( and ) are used to enter the large fraction into the one line entry box.

Tips on typing your problem into this software


Solving two equations with two unknowns

As you master you skills at solving one equation and one unknown, you may begin to study systems of equations, or simultaneous equations, involving two equations with two unknowns.  Perhaps in a more advanced math class, you’ll have even more than two equations to solve.

This selection will help you solve a system of two equations and two unknowns.  If you want to read a little bit more about equations and what they are, see this section entitled “Solving one equation and one unknown.”  With one equation you have a single equation containing a single unknown variable to solve for.  When solved using the rules of algebra, you don’t just get any answer, instead you arrive at a unique answer, that takes into account the properties, or characteristics of the single equation.  The equal sign that separates the left and right hand sides of an equation sets a very particular condition.  It means that no matter what, the left hand side must be equal to the right hand side.  This is why just any solution to the equation is not right; only the solution that preverses the delicate equality (the equal sign) between the left and right hand side will do!

With two equations and two unknowns, you have two variables to solve for.  Each equation has an equal sign, a left hand side, and a right hand side.  Now you will be solving for two variables, which are the two unknowns of the system.  But once again, not just any solution for the two unknown variables will do.  The two solutions are very specialized in that they too must preserve the delicate equality between the left and right hand sides of the equations; not just one or other other, but BOTH equations at the same time!  This is why systems of equations are sometimes called “simultaneous” equations.

So this selection will help you solve a system of two equations and two unknowns.  As you look at this window, notice there are two equal signs.  Each equal sign is on a separate line, and each equation in your problem goes on one of these lines or the other.

Notice also that each equal sign, has its own left hand side entry box, and right hand side entry box.  Those are the places where you should type each left hand side and right hand side of your equations.

After you enter your equations, you must tell the system what unknown variables you want to solve for.  These two variables go in the small boxes under the equation entry lines.  Once you have done this, just click on “Solve them!” to find your two solutions.

Here are some examples of what you may type here:

3x+y     =     12

x-y     =     7

What variables do you want to solve for? x and y

y     =     3x-1

3x+2y     =     16

What variables do you want to solve for? x and y

5m+2n     =     -8

4m+3n     =     2

What variables do you want to solve for? m and n

2a+2b     =     -3

5a+2b     =     6

What variables do you want to solve for? a and b

(2x+y)/3     =     15

(3x-y)/5     =     1

What variables do you want to solve for? x and y

This example will solve the two equations:

 and [image].  Notice how the parentheses are used to enter the large fractions into the single line equation box.

Tips on typing your problem into this software


Plotting or graphing on an XY or Carteisan Coordinate System

Being able to create a plot of an algebraic function is a good skill to learn, not only because it reenforces your understanding of the algebra you are working on, but because graphs (or plots) give visual information.  The old saying that “a picture is worth a thousand words” is definitely relevant here; graphs make pictures out of math.  Most students learn better visually than any other method, and typically pictures are easier to comprehend than an equation written down on a piece of paper.

This selection allows you to create a plot of any algebraic expression.  In an Algebra One class, most of the graphing done is for lines in point slope form (the form y=mx+b).  This selection can plot such lines quickly and easily.  As you move on in algebra, you might begin plotting parabolas, sine waves, cubic functions, etc.  This selection can handle all of these topics.  In some cases, you can even find the solution to two equations with two unknowns graphically.

The only condition is that you must manipulate the expression you need to plot so that it appears in the form of

y=some expression.

Notice that in this window, the “y=” part is already supplied for you.  You just need to type in the “some expression” part (see the examples given below).  If you need to use some variable in the “some expression” part, be sure the variable you use is “x.”  This selection only recognizes the letter “x” as a variable.  Note that “some expression” doesn’t have to contain a variable at all, if you are plotting some constant expression (e.g. a horizontal line).

You can also select the plot color you want your plot to be drawn in, and whether or not the plotting area should be cleared before drawing another plot.  Choosing not to clear the plotting area is good if you want to draw more than one plot in the graph window.

In the lower part of the window, there are four boxes allowing you to specify the minimum and maximum numbers that the x (horizontal) and y (vertical) should have.  For most plots you might make, the values that appear automatically should work just fine, but feel free to experiment!

In the lowest box, titled “How many points should be plotted?” you are to enter some number which tells the system how many points should be generated from your “y=”expression, and included on the graph.  If you’ve made graphs by hand before, usually you set up a little 2-column table labeled “x and y.”  You think of some point for x, and calculate what y will be, and fill in your table.  Perhaps you do this 3, 4, or 5 times, form a few points, then draw them on your graph.  Well the number that goes into this box is the number of times the computer will think of an x, find y, then draw the point on the graph.  If you make a graph and it looks really jagged and rough, try making this number higher.

Here are some examples of what you could plot:

•	To plot the line, y=5x+1, type 5x+1 into the “y=” box, then click “Plot it!”  Feel free to choose a plotting color, or change any of the other options.  Usually, the “y=” box is the only place you are required to type something.

•	To plot the line, y=-x, type -x into the “y=” box, then click “Plot it!”

•	To plot the line 3x+5y=1, you first have to do a little algebra on your equation, because this window requires your plot to be in the form “y=some expression.”  So what you would do first here, is solve this equation for y (or have this software do it for you!  See Solving one equation and one unknown).  If you solve this for y, you’ll get that y=(-3x+1)/5, or y=-3/5x+1/5.  You can type either one of these forms into the “y=” box in this window.  That is, type y=(-3x+1)/5, or y=-3/5x+1/5.  Now, click “Plot it!” and your line will be drawn.

•	To plot the parabola y=x2, enter x^2 into the “y=” box.

•	To plot the parabola y=(x-3)2, enter (x-3)^2 into the “y=” box.

•	To plot the sine wave, y=sin(5x), enter sin(5x) into the “y=” box.

•	To plot the curve y=cos2(x), enter cos(x)^2 into the “y=” box.

To solve the two equations x+2y=8 and x-y=2 graphically,

•	First, solve each equation for y, that is, get each equation into the required “y=” form.  The first becomes y=(-x+8)/2, and the second becomes y=x-2. (Or have this software do it for you!  See Solving one equation and one unknown).

•	Second, plot the first equation in a red color by typing (-x+8)/2 into the “y=” box.

•	Third, plot the second equation in a blue color by typing x-2 into the “y=” box.  And be sure to uncheck the “clear the plotting area…” option, so the first graph won’t be erased.

•	Now, look at the point on the graph where the red and blue lines cross.  This point is the (x,y) solution to these two equations!

Tips on typing your problem into this software


Creating Polar Plots

Polar plots are a somewhat more advanced method of generating plots.  Instead of the usual x-y coordinate system, polar plots are done on a more circular, radius-angle coordinate system.  The reason for producing polar plots is that for some types of curves, it’s much easier to describe them in a polar sense, than it is in an x-y sense.

Polar plots, as presented in the ususal mathematical form require two variables, the radius, r, and the angle, usually .  Since r is certainly available on a computer keyboard, but  is not, this selection uses r and t (t for theta), as the variables involved in producing polar plots.

You have to arrange your polar equation to plot in the form of

r=some expression that can involve t

Note that the “some expression” expression does not have to involve the variable “t” at all, in which case you would be plotting a curve that has a constant radius (e.g. a circle).  Also note that “t” is the only variable that this plotting selection will recognize; so if you need to use an independent variable, use “t.”

In creating  a polar plot, you type your expression to plot in the box next to the “r=” label.   The plot window itself already contains the “r=” part, so you don’t need to type this.

The two boxes below the “r=” box allow you to enter an angular range (in radians) through which the variable “t” should be passes in generating your plot.  The default setting of 0 to 6.28 will run “t” through a full circle (360 degrees).

You can also choose the plot color to use and whether or not the plotting area should be cleared before drawing the next plot.  Choosing not to clear the plotting area is helpful if you wish to draw two or more polar plots on the same graph.

You may also scale the plot by choosing a range for the polar radius.  This will effectively set what radial point should appear at the center of the plot (the minimum radius that goes in the left  box of the two), and what radial point should appear symmetrically around the edges of the plotting area (the maximum radius that goes in the right box of the two).

Lastly, you can choose the number of (r,) points that should be calculated by the software and included on your plot.  If your curve appears jagged and rough, try increasing this number.

Here are some examples of what you could type here:

Plot a circle with radius equal to 1, type

 r=1

The plot scale should include all points with a polar radius between:

0 and 2

To plot a spiral, type:

r=t

The polar variable t should run from:

 0 to 20

The plot scale should include all points with a polar radius between:

0 and 50

To plot a “star fish” type curve, type:

r=2+sin(5t)

The polar variable t should run from:

 0 to 6.28

The plot scale should include all points with a polar radius between:

0 and 5

To plot an ellipse type:

r=4/(4+1.2cos(2t))

The polar variable t should run from:

 0 to 6.28

The plot scale should include all points with a polar radius between:

0 and 10

To plot a “4 leaf” clover type:

r=cos(2t)

The polar variable t should run from:

 0 to 6.28

The plot scale should include all points with a polar radius between:

0 and 2


Plotting a Data File

At times, you may wish to create a plot from some raw data you have collected.  For example, you may record the temperature outside everyday for a week and want to see a graph of “temperature versus day.”  Or, you may watch the stock market, and might want to plot the price of your favorite stock versus day.

This selection will read a data file containing just such data from a file on your hard disk, and plot it.  You must, however, somehow create the file containing the data first.  There are a variety of way of doing this, as discussed below.

The file to plot must be in an ASCII (plain text, human readable) format.  Additionally, it must be a two column file, meaning the data points that are to represent the x, or horizontal axis of your plot must be in the first column, and the data points that correspond to each x-axis data point, that is to be plotted along the y-axis, must be in the second column.  This is all more complicated as it sounds.  Here’s an example.

Say you recorded the temperature outside at 12 noon everyday for 1 week (7 days).  Here is the data you collected.

Day #1: 62.3 degrees

Day #2: 65.5degrees

Day #3: 70.1 degrees

Day #4: 68 degrees

Day #5: 63.1 degrees

Day #6: 65.7 degrees

Day #7: 66.2 degrees

In order to make a plot of  “day versus temperature,” you must first put these data points in a file on your hard disk.  You can do this using Notepad or Wordpad, which come with every Windows-based computer.  You can also do it using a professional word processing system, just be sure to always save your file in “ASCII” or “plain text” format.

In this case, we’ll use Notepad.  We click the “Start” button, go to “Run..” and enter notepad in the box.  Into Notepad, we type in our data like this (two column format):

1 62.3

2 65.5

3 70.1

4 68

5 63.1

6 65.7

7 66.2

For this example, you can cut this data out of this help file and paste it into Notepad.  Be sure to put at least one space between each column, and press “enter” after each line.  When done, save the file to your hard disk.

Now, you return to the Webmath data plotting window.  Type the file name into the topmost box, or use the “Browse…” button to select it from your hard disk using the mouse.  When you have done so, just click “Plot it!” to plot your data file!  That’s it!

You may choose to autoscale the x and y axes, in which case the minimum and maximum values your two axes will have will be based on what numbers are in your data.  Alternatively, you can uncheck the “autoscale” box and enter your own axes values that will be used when producing the plot.


Solving Word Problems

(also known as verbal problems, story problems, etc.)

Students frequently have trouble solving word problems.  These are short, 1 to 4 sentence “stories” that outline some activity which raises some question.  The goal of these problems is for the student to use algebra to answer the question.

The question usually involves some unknown quantity like “how much” of something is needed, or “how long” will something take or “when” will something happen, etc. In order to answer any questions raised, or to “find the unknowns,” the student must translate the information given in the problem into concise algebraic equations.  And, although the frustrated student typically doesn’t think so, all information necessary to solve the problem is contained in the words and numbers given.

Once equations are formed, they may be solved in any usual manner.  The numerical solutions to these equations will answer the questions raised by the short story.  Usually, solving the equations is the easiest part of doing word problems.  The biggest problem a student faces is:

translating the words and numbers of the problem into algebraic equations that will solve the problem

As you use this section, remember the most famous 11 words muttered by the frustrated algebra student:

“I understand the concepts, I just can’t solve the word problem.”

This “story-to-equations” translation process is what the “Word Problems” module of this software tries to help you with.  We have compiled “a library” of more than 80 of the most common word problems found in today’s algebra books.  But rather than presenting the word problems “as is,” we have taken all of the specific information, (like names, numbers, measurements, etc.), out, and replaced them with “fill in boxes” and “list boxes” that you can type or click in the information that pertains to your specific word problem.  Once you do so, you may click “Solve it!,” to get a plainly worded, step by step solution to the word problem, using your own information.  At the end of a solution, Webmath will clearly tell you which equations need solving, but will not automatically solve them for you. You can either solve them yourself, or use the equation solving tools included in this software.  See Solving one equation and one unknown or Solving two equation and two unknowns for more information.

The word problem library is broken up into 6 different categories:

•	Ratio problems, which involve specific relationships between two or more numbers, like “a number is twice another and one-third of another.”

•	Age problems, which involve information on how the ages of people relate, like “Bill is twice as old as John.”

•	Number problems, which involve relationships between pure numbers, like “two consecutive numbers have a sum of 33.”

•	Rate problems, which involve traveling and the relationship that “distance=ratetime” or d=rt.  Trains leaving stations, people driving, and airplanes flying are common subjects for these types of problems.

•	Money problems, which involve how much money will be involve to buy something, or how much will be earned in a particular situation.  For example “Adult tickets cost $2.50, and children tickets cost $1.75…”

•	Mixture problems, which involve mixing things together.  Chemical mixtures (like 20% solutions of a medicine), or food (like walnuts at $1.75 per pound and almonds at $2.20 per pound) are common themes for problems like these.

There is a good chance that your exact problem is included in our library, but also a chance that it is not.  When using this section, look for a problem that is like the one you are working on.  Perhaps your problem deals with two airplanes traveling in some manner, but ours deals with two cars traveling in the same manner.  Yours might involve the ages of three bothers, ours might involve the ages of a parent and two children, with the same information.  Ours might deal with walnuts and almonds, yours might deal with pens and pencils. Remember that most of the time, this type of information, like specific names of people and places, types of objects (planes, cars, trains, walnuts, etc), or “feet, miles, pounds, liters” have no bearing on the overall logic that will solve a problem, so don’t focus too much of your attention on them.  They are there just to make the problem readable like a story.  Walnuts versus almonds, or pens versus pencils, or two types of candy might as well be called “Object 1” and “Object 2.”

Also, sometimes word problems offer more information or more words than are necessary to solve your problem, again, just to make the problems sound more like a complete story.  We have tried to take this “extra information” out of our library of problems and present only “bare bones” problems.  See if your problem can reduce down to the form of one in our library.

So be flexible as you look for help with your problem, and remember, that even a seemingly unrelated problem might give you a valuable hint to help you along with your solution.

When you find a problem you want to solve, click on each type-in box, or list-box presented to type or select information that you want to see the solution for.  Then click “Solve it!” and the solution will be presented.