Exercise 5 Page 264

When solving a system of equations using the Substitution Method, 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.

We are given the following system of equations. x+4y=6 & (I) x-y=1 & (II) Observing the given equations, it looks like it will be simplest to isolate x in the second equation.

x+4y=6 & (I) x-y=1 & (II)

AddEqn

(II): LHS+y=RHS+y

x+4y=6 & (I) x=1+y & (II)

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

x+4y=6 & (I) x=1+y & (II)

Substitute

(I):x= 1+y

1+y+4y=6 x=1+y

AddTerms

(I):Add terms

1+5y=6 x=1+y

SubEqn

(I):LHS-1=RHS-1

5y=5 x=1+y

DivEqn

(I):.LHS /5.=.RHS /5.

y=1 x=1+y

Now, to find the value of x, we need to substitute y=1 into either one of the equations in the given system. Let's use the second equation.

y=1 x=1+y

Substitute

(II):y= 1

y=1 x=1+ 1

AddTerms

(II): Add terms

y=1 x=2

The solution, or point of intersection, to this system of equations is the point (2, 1).

Checking Our Solution

To check our answer, we will substitute our solution into both equations. If doing so results in true statements, then our solution is correct.

x+4y=6 & (I) x-y=1 & (II)

(I), (II):x= 2, y= 1

2+4( 1)? =6 2- 1? =1

MultByOne

(I):a * 1=a

2+4? =6 2-1? =1

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

6=6 ✓ 1=1 ✓

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