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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 6 Page 207

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

( 43,- 56)

Practice makes perfect

In this system of equations, both equations have a term of 2y. Therefore, we will approach its solution with the Substitution Method. When solving a system of equations using this method, there are three steps.

Isolate a term in one of the equations.
Substitute the expression for that term into the other equation and solve for one of the variables.
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 2y in the second equation.

2x+2y=1 & (I) - x+2y=- 3 & (II)

AddEqn

(II): LHS+x=RHS+x

2x+2y=1 - x+2y+x=- 3+x

▼

(II): Add terms

CommutativePropAdd

(II): Commutative Property of Addition

2x+2y=1 - x+x+2y=x-3

AddTerms

(II): Add terms

2x+2y=1 2y=x-3

Now that we have isolated 2y, we can solve the system by substitution.

2x+2y=1 & (I) 2y=x-3 & (II)

Substitute

(I):2y= x-3

2x+( x-3)=1 2y=x-3

RemovePar

(I): Remove parentheses

2x+x-3=1 2y=x-3

AddTerms

(I): Add terms

3x-3=1 2y=x-3

AddEqn

(I): LHS+3=RHS+3

3x=4 2y=x-3

DivEqn

(I): .LHS /3.=.RHS /3.

3x3= 43 2y=x-3

▼

(I): Calculate quotient

MovePartNumRight

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

33x= 43 2y=x-3

QuotOne

(I): a/a=1

1x= 43 2y=x-3

IdPropMult

(I): Identity Property of Multiplication

x= 43 2y=x-3

Great! Now, to find the value of y, we need to substitute x= 43 into either one of the equations in the given system. Let's use the second equation.

x= 43 2y=x-3

Substitute

(II): x= 43

x= 43 2y= 43-3

NumberToFrac

(II): a = 3* a/3

x= 43 2y= 43- 93

SubFrac

(II): Subtract fractions

x= 43 2y=- 53

DivEqn

(II): .LHS /2.=.RHS /2.

x= 43 2y2= - 532

▼

(II): Calculate quotient

MovePartNumRight

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

x= 43 22y= - 532

QuotOne

(II): a/a=1

x= 43 1y= - 532

IdPropMult

(II): Identity Property of Multiplication

x= 43 y= - 532

MoveNegNumToFrac

(II): Put minus sign in front of fraction

x= 43 y=- 532

DivFracSL

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

x= 43 y=- 53(2)

Multiply

(II): Multiply

x= 43 y=- 56

The solution, or point of intersection, to this system of equations is the point ( 43,- 56).

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.

2x+2y=1 & (I) - x+2y=- 3 & (II)

(I), (II): x= 43, y= - 56

2( 43)+2( - 56)? =1 & (I) - 43+2( - 56)? =- 3 & (II)

▼

(I), (II): Simplify left-hand side

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

2( 43)-2( 56)? =1 & (I) - 43-2( 56)? =- 3 & (II)

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

2(4)3- 2(5)6? =1 & (I) - 43- 2(5)6? =- 3 & (II)

(I), (II): Multiply

83- 106? =1 & (I) - 43- 106? =- 3 & (II)

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

83- 53? =1 & (I) - 43- 53? =- 3 & (II)

(I), (II): Subtract fractions

33? =1 & (I) - 93? =- 3 & (II)

QuotOne

(I): a/a=1

1=1 & (I) - 93? =- 3 & (II)

CalcQuot

(II): Calculate quotient

1=1 ✓ - 3=- 3 ✓

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

Subchapter links

Try It p.206-207

Self-Assessment for Concepts & Skills p.207

Self-Assessment for Problem Solving p.208

Review & Refresh p.209

Concepts, Skills, & Problem Solving p.209-210

Exercise 6 Page 207

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Practice exercises

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Checking Our Answer