Exercise 22 Page 375

When solving a system of equations using substitution, 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.

Consider the given equations, we need to isolate one of the variables. Let's start by isolating x in Equation I.

4y-x=5+2y & (I) 3x+7y=24 & (II)

AddEqn

(I): LHS+x=RHS+x

4y=5+2y+x 3x+7y=24

SubEqn

(I): LHS-2y=RHS-2y

2y=5+x 3x+7y=24

SubEqn

(I): LHS-5=RHS-5

2y-5=x 3x+7y=24

RearrangeEqn

(I): Rearrange equation

x=2y-5 3x+7y=24

Now that we've isolated x, we can solve the system by substitution.

x=2y-5 & (I) 3x+7y=24 & (II)

Substitute

(II):x= 2y-5

x=2y-5 3( 2y-5)+7y=24

Distr

(II):Distribute 3

x=2y-5 6y-15+7y=24

AddTerms

(II):Add terms

x=2y-5 13y-15=24

AddEqn

(II):LHS+15=RHS+15

x=2y-5 13y=39

DivEqn

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

x=2y-5 y=3

Now, to find the value of x, we need to substitute y=3 into either one of the equations in the given system. Let's use the first equation.

x=2y-5 y=3

Substitute

(I):y= 3

x=2( 3)-5 y=3

Multiply

(I): Multiply

x=6-5 y=3

SubTerm

(I): Subtract term

x=1 y=3

The solution, or point of intersection, to this system of equations is the point (1,3).

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.

4y-x=5+2y & (I) 3x+7y=24 & (II)

(I), (II): x= 1, y= 3

4( 3)- 1? =5+2( 3) 3( 1)+7( 3)? =24

(I), (II): Multiply

12-1? =5+6 3+21? =24

(I), (II): Add and subtract terms

11=11 24=24

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