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CHAPTER8.3T
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à 8.3ïSolving Equations by Substitution.
äèPlease solve the following systems of equations by the
êê Substitution Method.
âêèx + 2y = 4êêë3x - y = 5
#êïSolve,ï3x - y = 5.ï┌─¥Sub. inë3(4-2y)-y = 5èx = 4 - 2y
#êêêêï│ïthe secondï12 - 6y-y = 5èx = 4 - 2∙1
#Solve the firstêê │ïequation.è12 - 7y = 5èºïx = 4 - 2
#equation for x.ïx + 2y = 4è│êêï-7y = -7è │è x = 2
#êêèx = 4 - 2y ──┘êêè y = 1ï───┘
êêêêêë The solution is (2,1).
éSêêë x + 2y = 4
ëTo solve the system,ï3x - y = 5 , by the Substitution Method,
you should solve the first equation for x.
êêêë x + 2y = 4
êêêë x = 4 - 2y
The expression, 4 - 2y, is then substituted for x in the second
equation.
êêêë 3x - y = 5
êêêè3(4 - 2y) - y = 5
êêêè 12 - 6y - y = 5
êêêë12 - 7y = 5
êêêê-7y = -7
êêêêïy = 1
The "1" is substituted in for y in either of the original equations
to find the value of x.
êêêêx = 4 - 2y
êêêë x = 4 - 2(1)
êêêêx = 4 - 2
êêêêïx = 2
The solution of the system of equations is the ordered pair, (2,1).
When solving a system of equations by the Substitution Method, you
can solve for either the "x" variable, as we did in this example,
or for the "y" variable.ïYou would choose the variable that involves
the least amount of work to isolate.
1
êêêë2x - 3y = 3
ë Solve the system,è-x + 4y = 1ï, by the Substitution Method.
êë A)ï(3,1)ëB)ï(6,3)ëC)ï(7,2)ëD)ïå
ü
êêêSolve,ï2x - 3y = 3
êêêê -x + 4y = 1.
êêèSolve the second equation for x.
êêêê-x + 4y = 1
êêêê-x = -4y + 1
êêêê x = 4y - 1
ëSubstitute the expression, 4y -1, in for x in the first equation.
êêêê 2x - 3y = 3
#êêêë2(4y - 1) - 3y = 3ï┌─¥è x = 4y - 1
#êêêê8y - 2 - 3y = 3è│ë x = 4(1) - 1
#êêêêè5y - 2 = 3ë│ë x = 4 - 1
#êêêêë5y = 5ê│ë x = 3
#êêêêë y = 1ï─────┘
The solution of the system of equations is the ordered pair, (3,1).
ÇïA
2
êêêë3x + 2y = 5
ë Solve the system,è-2x + y = 6ï, by the Substitution Method.
êè A)ï(-2,2)ëB)ï(3,-2)ëC)ï(-1,4)ëD)ïå
ü
êêêSolve,ï3x + 2y = 5
êêêê -2x + y = 6.
êêèSolve the second equation for y.
êêêê-2x + y = 6
êêêê y = 2x + 6
ëSubstitute the expression, 2x + 6, in for y in the first equation.
êêêê 3x + 2y = 5
#êêêë3x + 2(2x + 6) = 5ï┌─¥è y = 2x + 6
#êêêë 3x + 4x + 12 = 5è│ë y = 2(-1) + 6
#êêêêï7x + 12 = 5ë│ë y = -2 + 6
#êêêêë7x = -7ë │ë y = 4
#êêêêëx = -1ï─────┘
The solution of the system of equations is the ordered pair, (-1,4).
ÇïC
3
êêêëx - 4y = -1
ë Solve the system,è2x + y = 16ï, by the Substitution Method.
êè A)ï(3,1)ëB)ï(7,2)ëC)ï(8,0)ëD)ïå
ü
êêêSolve,ïx - 4y = -1
êêêê 2x + y = 16.
êêèSolve the first equation for x.
êêêê x - 4y = -1
êêêê x = 4y - 1
è Substitute the expression, 4y - 1, in for x in the second equation.
êêêê 2x + y = 16
#êêêë2(4y - 1) + y = 16ï┌─¥è x = 4y - 1
#êêêë 8y - 2 + y = 16è │ë x = 4(2) - 1
#êêêê 9y - 2 = 16ë │ë x = 8 - 1
#êêêêë9y = 18ë │ë x = 7
#êêêêë y = 2ï─────┘
The solution of the system of equations is the ordered pair, (7,2).
ÇïB
4
êêêë3x + 2y = -8
ë Solve the system,è5x - y = 4è, by the Substitution Method.
êè A)ï(0,-4)ëB)ï(1,1)ëC)ï(-2,-1)ëD)ïå
ü
êêêSolve,ï3x + 2y = -8
êêêêï5x - y = 4ï.
êêèSolve the second equation for y.
êêêêï5x - y = 4
êêêê -y = -5x + 4
êêêêïy = 5x - 4
è Substitute the expression, 5x - 4, in for y in the first equation.
êêêê 3x + 2y = -8
#êêêë3x + 2(5x - 4) = -8 ┌─¥è y = 5x - 4
#êêêë 3x + 10x - 8 = -8ï│ë y = 5(0) - 4
#êêêê 13x - 8 = - 8è │ë y = 0 - 4
#êêêêè 13x = 0ê│ë y = -4
#êêêêë x = 0ï─────┘
The solution of the system of equations is the ordered pair, (0,-4).
ÇïA
5
êêêë2x = 3y + 7
ë Solve the system,è4y = x - 1è, by the Substitution Method.
êè A)ï(9,2)ëB)ï(8,3)ëC)ï(5,1)ëD)ïå
ü
êêêSolve,ï2x = 3y + 7
êêêê 4y = x - 1è.
êêèSolve the second equation for x.
êêêêï4y = x - 1
êêêêï4y + 1 = x
êêêêïx = 4y + 1
è Substitute the expression, 4y + 1, in for x in the first equation.
êêêê 2x = 3y + 7
#êêêë2(4y + 1) = 3y + 7ï┌─¥è x = 4y + 1
#êêêë 8y + 2 = 3y + 7è │ë x = 4(1) + 1
#êêêêï5y + 2 = 7ë │ë x = 4 + 1
#êêêêë5y = 5ê│ë x = 5
#êêêêë y = 1ï─────┘
The solution of the system of equations is the ordered pair, (5,1).
ÇïC
6
êêêë3x = 21 - 5y
ë Solve the system,è-4x + 3y = 1 , by the Substitution Method.
êè A)ï(7,0)ëB)ï(-4,-5)ëC)ï(2,3)ëD)ïå
ü
êêêSolve,ï3x = 21 - 5y
êêêê -4x + 3y = 1 .
êêèSolve the first equation for x.
êêêêï3x = 21 - 5y
êêêêïx = 7 - (5/3)y
èSubstitute the expression, 7-5/3y, in for x in the second equation.
êêêê -4x + 3y = 1
#êêêï-4(7-(5/3)y) + 3y = 1ï┌─¥è x = 7 - (5/3)y
#êêêï-28 + (20/3)y + 3y =1ï│ë x = 7 - (5/3)∙3
#êêêè -84 + 20y + 9y = 3è│ë x = 7 - 5
#êêêêè 29y = 87ë │ë x = 2
#êêêêë y = 3ï─────┘
The solution of the system of equations is the ordered pair, (2,3).
ÇïC