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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
2. Solving Systems of Linear Equations by Substitution
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Exercise 9 Page 209

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

(- 2,- 92)

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.

For this exercise, x is already isolated in one equation, so we can skip straight to solving!

x=2y+7 & (I) 3x-2y=3 & (II)
Substitute

(II):x= 2y+7

x=2y+7 3( 2y+7)-2y=3
Distr

(II): Distribute 3

x=2y+7 3(2y)+3(7)-2y=3
â–¼
(II): Simplify left-hand side
Multiply

(II): Multiply

x=2y+7 6y+21-2y=3
CommutativePropAdd

(II): Commutative Property of Addition

x=2y+7 6y-2y+21=3
SubTerm

(II): Subtract term

x=2y+7 4y+21=3
SubEqn

(II): LHS-21=RHS-21

x=2y+7 4y+21-21=3-21
SubTerm

(II): Subtract term

x=2y+7 4y=- 18

Now, we can use the Division Property of Equality to simplify the equation even further.

x=2y+7 4y=- 18
DivEqn

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

x=2y+7 4y4= - 184
â–¼
Simplify left-hand side
MovePartNumRight

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

x=2y+7 44y= - 184
CalcQuot

(II): Calculate quotient

x=2y+7 1y= - 184
IdPropMult

(II): Identity Property of Multiplication

x=2y+7 y= - 184
â–¼
Simplify right-hand side
MoveNegNumToFrac

(II): Put minus sign in front of fraction

x=2y+7 y=- 184
ReduceFrac

(II): a/b=.a /2./.b /2.

x=2y+7 y=- 92

Great! Now, to find the value of x, we need to substitute y=- 92 into the first equation.

x=2y+7 y=- 92
Substitute

(I): y= - 9/2

x=2( - 92)+7 y=- 92
MoveNegFracToNum

(I): Put minus sign in numerator

x=2( - 92)+7 y=- 92
DenomMultFracToNumber

(I): 2 * a/2= a

x=- 9+7 y=- 92
AddTerms

(I): Add terms

x=- 2 y=- 92

The solution, or point of intersection, to this system of equations is the point (- 2,- 92).

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=2y+7 & (I) 3x-2y=3 & (II)

(I), (II): x= - 2, y= - 92

- 2? =2( - 92)+7 3( - 2)-2( - 92)? =3

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

- 2? =-2( 92)+7 -3( 2)-2(- 92)? =3
MultNegNegOnePar

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

- 2? =-2( 92)+7 -3( 2)+2( 92)? =3

(I), (II): 2 * a/2= a

- 2? =-9+7 -3( 2)+9? =3
Multiply

(I): Multiply

- 2? =-9+7 -6+9? =3

(I), (II): Add terms

- 2=- 2 ✓ 3=3 ✓

Because both equations are true statements, we know that our solution is correct.

Subchapter links expand_more
arrow_right Try It p.206-207
1
Try It 2 (Page 206)
Try It 3 (Page 206)
Try It 4 (Page 207)
Try It 5 (Page 207)
Try It 6 (Page 207)
arrow_right Self-Assessment for Concepts & Skills p.207
Self-Assessment for Concepts & Skills 7 (Page 207)
Self-Assessment for Concepts & Skills 8 (Page 207)
Self-Assessment for Concepts & Skills 9 (Page 207)
Self-Assessment for Concepts & Skills 10 (Page 207)
Self-Assessment for Concepts & Skills 11 (Page 207)
Self-Assessment for Concepts & Skills 12 (Page 207)
Self-Assessment for Concepts & Skills 13 (Page 207)
arrow_right Self-Assessment for Problem Solving p.208
Self-Assessment for Problem Solving 14 (Page 208)
Self-Assessment for Problem Solving 15 (Page 208)
arrow_right Review & Refresh p.209
Review & Refresh 1 (Page 209)
Review & Refresh 2 (Page 209)
Review & Refresh 3 (Page 209)
Review & Refresh 4 (Page 209)
arrow_right Concepts, Skills, & Problem Solving p.209-210
5
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Concepts, Skills, & Problem Solving 7 (Page 209)
Concepts, Skills, & Problem Solving 8 (Page 209)
Concepts, Skills, & Problem Solving 9 (Page 209)
Concepts, Skills, & Problem Solving 10 (Page 209)
Concepts, Skills, & Problem Solving 11 (Page 209)
Concepts, Skills, & Problem Solving 12 (Page 209)
Concepts, Skills, & Problem Solving 13 (Page 209)
Concepts, Skills, & Problem Solving 14 (Page 209)
Concepts, Skills, & Problem Solving 15 (Page 209)
Concepts, Skills, & Problem Solving 16 (Page 209)
Concepts, Skills, & Problem Solving 17 (Page 210)
Concepts, Skills, & Problem Solving 18 (Page 210)
Concepts, Skills, & Problem Solving 19 (Page 210)
Concepts, Skills, & Problem Solving 20 (Page 210)
Concepts, Skills, & Problem Solving 21 (Page 210)
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Concepts, Skills, & Problem Solving 23 (Page 210)
Concepts, Skills, & Problem Solving 24 (Page 210)
Concepts, Skills, & Problem Solving 25 (Page 210)
Concepts, Skills, & Problem Solving 26 (Page 210)