Exercise 16 Page 375

When solving a system of equations using substitution, there are three steps.

Isolate a variable in one of the equations.
Substitute the expression for that variable into the other equation and solve.
Substitute this solution into one of the equations and solve for the value of the other variable.

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

3x+2y=23 & (I) 12x-4=y & (II)

Substitute

(I):y= 12x-4

3x+2( 12x-4)=23 12x-4=y

Distr

(I):Distribute 2

3x+x-8=23 12x-4=y

AddTerms

(I):Add terms

4x-8=23 12x-4=y

AddEqn

(I):LHS+8=RHS+8

4x=31 12x-4=y

DivEqn

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

x= 314 12x-4=y

Now, to find the value of y, we need to substitute x= 314 into either one of the equations in the given system. Let's use the second equation.

x= 314 12x-4=y

Substitute

(II):x= 314

x= 314 12( 314)-4=y

MultFrac

(II):Multiply fractions

x= 314 318-4=y

NumberToFrac

(II): a = 8* a/8

x= 314 318- 328=y

SubFrac

(II): Subtract fractions

x= 314 - 18=y

RearrangeEqn

(II): Rearrange equation

x= 314 y=- 18

The solution, or point of intersection, to this system of equations is the point ( 314,- 18).

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.

3x+2y=23 & (I) 12x-4=y & (II)

(I), (II): x= 31/4, y= -1/8

3( 314)+2( - 18)? =23 12( 314)-4? = - 18

(I), (II): Multiply

934- 28? =23 318-4? =- 18

ReduceFrac

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

934- 14? =23 318-4? =- 18

NumberToFrac

(II): a = 8* a/8

934- 28? =23 318- 328? =- 18

(I), (II): Subtract fractions

924? =23 - 18=- 18

CalcQuot

(I): Calculate quotient

23=23 - 18=- 18

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

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