Exercise 5 Page 213

To solve a system of linear equations using the Elimination Method, one of the variable terms needs to be eliminated when one equation is added to or subtracted from the other. This means that either the x- or the y-terms must cancel each other out. 4 x-5 y=-19 & (I) - x-2 y=8 & (II) Currently, none of the terms in this system will cancel out. Therefore, we need to find a common multiple between two variable like terms in the system. If we multiply Equation (II) by 4 the x-terms will have opposite coefficients. 4 x-5 y=-19 4(- x-2 y)=4(8) ⇓ 4x-5 y=-19 -4x-8 y=32 We can see that the x-terms will eliminate each other if we add Equation (I) to Equation (II).

4x-5y=-19 -4x-8y=32

SysEqnAdd

(II): Add (I)

4x-5y=-19 -4x-8y+( 4x-5y)=32+( -19)

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(II):Solve for y

RemovePar

(II): Remove parentheses

4x-5y=-19 -4x-8y+4x-5y=32+(-19)

AddNeg

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

4x-5y=-19 -4x-8y+4x-5y=32-19

AddSubTerms

(II): Add and subtract terms

4x-5y=-19 -13y=13

DivEqn

(II): .LHS /(-13).=.RHS /(-13).

4x-5y=-19 -13y/-13=13/-13

MovePartNumRight

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

4x-5y=-19 -13/-13 * y=13/-13

DivNegNeg

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

4x-5y=-19 13/13 * y=13/-13

MoveNegDenomToFrac

(II): Put minus sign in front of fraction

4x-5y=-19 13/13 * y=-13/13

QuotOne

(I): a/a=1

4x-5y=-19 1 * y=-1

IdPropMult

(I): Identity Property of Multiplication

4x-5y=-19 y=-1

Now, we can solve for x by substituting the value of y into either equation and simplifying.

4x-5y=-19 y=-1

Substitute

(I): y= -1

4x-5( -1)=-19 y=-1

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(I):Solve for x

MultPropNegOne

(I): Multiplication Property of -1

4x-(-5)=-19 y=-1

SubNeg

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

4x+5=-19 y=-1

SubEqn

(I): LHS-5=RHS-5

4x+5-5=-19-5 y=-1

SubTerms

(I): Subtract terms

4x=-24 y=-1

DivEqn

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

4x/4=-24/4 y=-1

MovePartNumRight

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

4/4 * x=-24/4 y=-1

QuotOne

(I): a/a=1

1 * x=-24/4 y=-1

IdPropMult

(I): Identity Property of Multiplication

x=-24/4 y=-1

MoveNegNumToFrac

(I): Put minus sign in front of fraction

x=-24/4 y=-1

CalcQuot

(I): Calculate quotient

x=-6 y=-1

The solution, or point of intersection, of the system of equations is (-6,-1). To check this solution, we will substitute it back into the given system and simplify. If doing so results in true statements for every equation in the system, our solution is correct.

4x-5y=-19 & (I) - x-2y=8 & (II)

(I), (II): x= -6, y= -1

4( -6)-5( -1) ? = -19 -( -6)-2( -1) ? = 8

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Simplify

MultPosNeg

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

-24-5(-1) ? = -19 -(-6)-2(-1) ? = 8

NegNeg

(II): - (- a)=a

-24-5(-1) ? = -19 6-2(-1) ? = 8

(I), (II): Multiplication Property of -1

-24-(-5) ? = -19 6-(-2) ? = 8

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

-24+5 ? = -19 6+2 ? = 8

(I), (II): Add terms

-19 = -19 ✓ & (I) 8 = 8 ✓ & (II)