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CHAPTER3.1T
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à 3.1ïEquations and the addition property
äïPlease solve the following equations by using the addition
property.
âS
êëx - 5 = 3êêêx + 7 = 4
êëx - 5 + 5 = 3 + 5êë x + 7 + (-7) = 4 + (-7)
êëx + 0 = 8êêêxï+ 0 = -3
êëx = 8êêêè x = -3
éS
The addition property of equality simply states that you may add the
same number to both sides of an existing equation.ïWhen solving an
equation such as, x - 6 = 9, the goal is to get "x" by itself on one
side of the equation or the other.ïThe number on the opposite side is
then the solution to the original equation.ïThe solution is the number
that makes a true sentence out of the original equation.
The addition principle can be used to isolate "x".ïIf 6 is added to
both sides of the equationïx - 6 = 9,ïyou getïx - 6 + 6 = 9 + 6. The
-6 and +6 cancel to zero since they are additive inverses.ïThus
x + 0 = 15 is the solution.
1
êêêë Solveïx - 7 = 3
ëA)ï4êè B)ï-4êè C)ï10ê d) å of ç
ü
êêêê x - 7 = 3
êêêê x - 7 + 7 = 3 + 7
êêêê x + 0 = 10
êêêê x = 10
Ç C
2
êêêë Solveïx + 8 = 5
ëA)ï-3êèB)ï13êè C)ï-13êd) å of ç
ü
êêêê x + 8 = 15
êêêê x + 8 + (-8) = 5 + (-8)
êêêê x + 0 = -3
êêêê x = -3
Ç A
3
êêêèSolveï6x + 5 = 5x - 2
ëA)ï-3êèB)ï-7êè C)ï10ê d) å of ç
ü
êêêê6x + 5 = 5x - 2
êêêê6x + (-5x) + 5 = 5x + (-5x) - 2
êêêêx + 5 = -2
êêêêx + 5 + (-5) = -2 + (-5)
êêêêx = -7
Ç B
4
êêêïSolveï-3y + 10 = -4y - 8
ëA)ï-8êèB)ï10êè C)ï-18êd) å of ç
ü
êêêê-3y + 10 = -4y - 8
êêêê-3y + 4y + 10 = -4y + 4y - 8
êêêêy + 10 = 0 - 8
êêêêy + 10 - 10 = -8 - 10
êêêêy = -18
Ç C
5êêSolveè5ë1
êêêêè─∙z = ─∙z - 7
êêêêè4ë4
êêêè 3
ëA)ï-7êèB)ï─∙zêè C)ï-1ê d) å of ç
êêêè 4
üë 5ë1êê5ë┌ï1ï┐è1ë┌ï1ï┐
êë─∙z = ─∙z - 7ê ─∙z + │- ─∙z│ = ─∙z + │- ─∙z│ - 7
êë4ë4êê4ë└ï4ï┘è4ë└ï4ï┘
êë4
êë─∙z = 0 - 7ê z = 0 - 7êïz = -7
êë4
Ç A
6
êêë Solveï3a + 2 + 2a - 4a = 4 + 11
ëA)ï-6êèB)ï13êè C)ï9êïd) å of ç
üêêë3a + 2 + 2a - 4a = 4 + 11
êêêè (3a + 2a - 4a) + 2 = 4 + 11
êêêè (a) + 2 = 15
êêêè a + 2 - 2 = 15 -2
êêêè a + 0 = 13
êêêè a = 13
Ç B
7
êêïSolve 6k + 3 - 14k + 2k - 8 = -7k + 4
ëA)ï4êè B)ï-6êè C)ï9êïd) å of ç
üêêè 6k + 3 - 14k + 2k - 8 = -7k + 4
êêêè(6k - 14k + 2k) + (3 - 8) = -7k + 4
êêêè-6k + (-5) = -7k + 4
êêêè-6k + 7k + (-5) = -7k + 7k + 4
êêêèk + (-5) = 0 + 4
êêêèk + (-5) + 5 = 4 + 5
êêêèk = 9
Ç C
8
êêë Solveï-2(x + 3) + 3(x - 6) =4
ëA)ï28êèB)ï-6êè C)ï14ê d) å of ç
üêêë -2(x + 3) + 3(x - 6) = 4
êêêë-2x - 6 + 3x - 18 = 4
êêêë(-2x + 3x) + (-6 - 18) = 4
êêêëx + (-24) = 4
êêêëx + (-24) + 24 = 4 + 24
êêêëx + 0 = 28
êêêëx = 28
Ç A
9
êêëSolveï-6(2k - 4) + (2 + 13k) = 5
ëA)ï14êèB)ï-21êèC)ï6êïd) å of ç
üêêë -6(2k - 4) + (2 + 13k) = 5
êêêë-12k + 24 + 2 + 13k = 5
êêêë(-12k + 13k) + (24 + 2) = 5
êêêëk + 26 = 5
êêêëk + 26 + (-26) = 5 + (-26)
êêêëk + 0 = -21
êêêëk = -21
Ç B
äïPlease identify the following equations as conditional,
identity, or inconsistent.
âSèx - 6 = 7êïx - 6 = x - 6êèx + 3 = x + 4
êè x-6 + 6 = 7 + 6èx+(-x)-6 = x+(-x)-6è x+(-3)+3 = x+(-x)+4
êè x + 0 = 13ê 0 - 6 = 0 - 6êè0 + 3 = 0 + 4
êè x = 13êë-6 = -6êêï3 = 4
êè conditionalêidentityêê inconsistent
êè solution is 13è solution is allê no solution
êêêê real numbers
éS
First degree equations that have one solution are called conditional
equations.ïThere is just one number that makes the original equation
a true sentence.
Some equations though are true no matter what number you substitute in
for "x".ïThese are called identities and the solution set is all of the
real numbers.
The last type of linear equation is false no matter what number you
substitute in for "x".ïThis type is called inconsistent and has no
solution.
x + 6 = 9êêë x + 6 = x + 6êê x + 6 = x + 7
conditionalêêè identityêêë inconsistent
10
êêèWhich type of equation is x - 6 = 14?
A) conditionalëB)ïidentityëC) inconsistentëD) å of ç
ü
êêêêïx - 6 = 14
êêêêïx - 6 + 6 = 14 + 6
êêêêïx + 0 = 20
êêêêïx = 20
êêëone solution implies conditional
Ç A
11
êêêèWhich type of equation is
êêê 5k + 4 - 12k + 4k - 8 = -3k +2
A) conditionalëB)ïidentityëC) inconsistentëD) å of ç
üêêï5k + 4 - 12k + 4k - 8 = -3k + 2
êêê (5k - 12k + 4k) + (4 - 8) = -3k + 2
êêê -3k + (-4) = -3k + 2
êêê -3k + 3k + (-4) = -3k + 3k + 2
êêê 0 + (-4) = 0 + 2
êêê -4 = 2
êêê inconsistent, no solution
Ç C
12
êêêïWhich type of equation is
êêêë 2(x + 3) = 2x + 6
A) conditionalëB)ïidentityëC) inconsistentëD) å of ç
üêêê2(x + 3) = 2x + 6
êêêë 2x + 6 = 2x + 6
êêêë 2x - 2x + 6 = 2x - 2x + 6
êêêë 0 + 6 = 0 + 6
êêêë 6 = 6
êê identity, solution is all real numbers
Ç B
13
êêèSolveï5(-3x + 2) = -6(x - 2) - 9x - 2
A) 6ë B) all real numbersëC) no solutionë D) å of ç
ü
êêêï5(-3x + 2) = -6(x - 2) - 9x - 2
êêê -15x + 10 = -6x + 12 - 9x - 2
êêê -15x + 10 = -15x + 10
êêêê 10 = 10
êêêïidentity, all real numbers
Ç B