Exercise 12 Page 393

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 equation. In this exercise, this means that either the x-terms or the y-terms must cancel each other out. 3 x+6 y=42 & (I) -7 x+8 y=-109 & (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 (I) by 7 and multiply (II) by 3, the x-terms will have opposite coefficients. 7(3 x+6 y)=7(42) 3(-7 x+8 y)=3(-109) ⇒ 21x+42 y=294 - 21x+24 y=-327We can see that the x-terms will eliminate each other if we add (I) to (II).

21x+42y=294 -21x+24y=-327

SysEqnAdd

(II): Add (I)

21x+42y=294 -21x+24y+ 21x+42y=-327+ 294

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(II):Solve for y

AddSubTerms

(II): Add and subtract terms

21x+42y=294 66y=-33

DivEqn

(II): .LHS /66.=.RHS /66.

21x+42y=294 y=- 3366

ReduceFrac

(II): a/b=.a /33./.b /33.

21x+42y=294 y=- 12

Now, we can now solve for y by substituting the value of x into either equation and simplifying. Let's use the first equation.

21x+42y=294 y=- 12

Substitute

(I): y= -1/2

21x+42( - 12)=294 y=- 12

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(I):Solve for x

MoveNegFracToNum

(I):Put minus sign in numerator

21x+42( -12)=294 y=- 12

MoveLeftFacToNum

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

21x+ 42(-1)2=294 y=- 12

MultPropNegOne

(I): Multiplication Property of -1

21x+ (-42)2=294 y=- 12

CalcQuot

(I): Calculate quotient

21x+(-21)=294 y=- 12

AddNeg

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

21x-21=294 y=- 12

AddEqn

(I): LHS+21=RHS+21

21x=315 y=- 12

DivEqn

(I): .LHS /21.=.RHS /21.

x=15 y=- 12

The solution, or point of intersection, of the system of equations is (15,- 12).