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Big Ideas Math: Modeling Real Life, Grade 8
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Big Ideas Math: Modeling Real Life, Grade 8 View details
3. Solving Systems of Linear Equations by Elimination
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Exercise 2 Page 216

Does either of the equations have an isolated variable in it?

(-9,-2)

Practice makes perfect

When solving a system of equations using the Substitution Method, there are three steps.

  1. Isolate a variable in one of the equations.
  2. Substitute the expression for that variable into the other equation and solve.
  3. Substitute this solution into one of the equations and solve for the value of the other variable.
Observing the given equations, it looks like it will be simplest to isolate x in the first equation using the Properties of Equality.
x-5y=1 & (I) - x+y=7 & (II)
AddEqn

(I): LHS+5y=RHS+5y

x-5y+5y=1+5y - x+y=7
AddTerms

(I): Add terms

x=1+5y - x+y=7
Now that we've isolated x, we can solve the system by substitution.
x=1+5y - x+y=7
Substitute

(II):x= 1+5y

x=1+5y - ( 1+5y)+y=7
â–Ľ
(II):Solve for y
Distr

(II):Distribute (-1)

x=1+5y (-1)1+(-1)5y+y=7
MultPropNegOne

(II):Multiplication Property of -1

x=1+5y -1+(-5y)+y=7
AddNeg

(II):a+(- b)=a-b

x=1+5y -1-5y+y=7
AddEqn

(II): LHS+1=RHS+1

x=1+5y -1-5y+y+1=7+1
AddTerms

(II):Add terms

x=1+5y -4y=8
DivEqn

(II):.LHS /(-4).=.RHS /(-4).

x=1+5y -4y/-4=8/-4
MovePartNumRight

(II): a* b/c=a/c* b

x=1+5y -4/-4 * y=8/-4
DivNegNeg

(II): - a/- b=a/b

x=1+5y 4/4 * y=8/-4
QuotOne

(II): a/a=1

x=1+5y 1 * y=8/-4
IdPropMult

(II): Identity Property of Multiplication

x=1+5y y=8/-4
MoveNegDenomToFrac

(II): Put minus sign in front of fraction

x=1+5y y=-8/4
CalcQuot

(II): Calculate quotient

x=1+5y y=-2
Great! Now, to find the value of x, we need to substitute y=-2 into either one of the equations in the given system. Let's use the first equation.
x=1+5y y=-2
Substitute

(I):y= -2

x=1+5( -2) y=-2
â–Ľ
Solve for x
MultPosNeg

(I):a(- b)=- a * b

x=1+( -10) y=-2
AddNeg

(I):a+(- b)=a-b

x=1-10 y=-2
SubTerms

(I): Subtract terms

x=-9 y=-2
The solution, or point of intersection, to this system of equations is the point (-9,-2).

Checking Our Answer

To check our answer, we will substitute our solution into both equations. If doing so results in true statements, then our solution is correct.
x-5y=1 & (I) - x+y=7 & (II)

(I), (II): x= -9, y= -2

-9-5( -2) ? = 1 -( -9)+( -2) ? = 7
â–Ľ
Simplify
MultPosNeg

(I): a(- b)=- a * b

-9-(-10) ? = 1 -(-9)+(-2) ? = 7
SubNeg

(I): a-(- b)=a+b

-9+10 ? = 1 -(-9)+(-2) ? = 7
NegNeg

(II): - (- a)=a

-9+10 ? = 1 9+(-2) ? = 7
AddNeg

(II): a+(- b)=a-b

-9+10 ? = 1 9-2 ? = 7

(I), (II): Add and subtract terms

1 = 1 âś“ 7 = 7 âś“
Because both equations are true statements, we know that our solution is correct.
Subchapter links expand_more
arrow_right Try It p.212-214
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Try It 2 (Page 212)
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arrow_right Self-Assessment for Concepts & Skills p.214
Self-Assessment for Concepts & Skills 8 (Page 214)
Self-Assessment for Concepts & Skills 9 (Page 214)
Self-Assessment for Concepts & Skills 10 (Page 214)
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Self-Assessment for Concepts & Skills 12 (Page 214)
Self-Assessment for Concepts & Skills 13 (Page 214)
Self-Assessment for Concepts & Skills 14 (Page 214)
arrow_right Self-Assessment for Problem Solving p.215
Self-Assessment for Problem Solving 15 (Page 215)
Self-Assessment for Problem Solving 16 (Page 215)
arrow_right Review & Refresh p.216
Review & Refresh 1 (Page 216)
Review & Refresh 2 (Page 216)
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Review & Refresh 4 (Page 216)
Review & Refresh 5 (Page 216)
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Concepts, Skills, & Problem Solving 9 (Page 216)
Concepts, Skills, & Problem Solving 10 (Page 216)
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Concepts, Skills, & Problem Solving 14 (Page 216)
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Concepts, Skills, & Problem Solving 16 (Page 216)
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Concepts, Skills, & Problem Solving 18 (Page 217)
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