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Solving Linear Systems of Equations by Substitution

Example

Use substitution to find the solution of this system.

-3x + 4y

x - 6y

= 17

= -8

First equation

Second equation

Solution

Step 1 Solve one equation for one of the variables in terms of the other variable.

Either equation may be solved for either variable.

For instance, let’s solve the second equation for x.

Add 6y to both sides.

 

x - 6y

x

 

= -8

= 6y - 8

Step 2 Substitute the expression found in Step 1 into the other equation. Then, solve for the variable.

 

Substitute 6y - 8 for x in the first equation.

Remove parentheses.

Combine like terms.

Subtract 24 from both sides.

Divide both sides by -14.

Now we know .

Next, we will find x.

 -3x + 4y

-3(6y - 8) + 4y

 -18y + 24 + 4y

-14y + 24

-14y

y

= 17

= 17

= 17

= 17

= -7

 

Step 3 Substitute the value obtained in Step 2 into one of the equations containing both variables. Then, solve for the remaining variable.

We will use the equation from Step 1. x = 6y - 8
Substitute for y.

x

Simplify.

The solution of the system is

x

= -5
 

Step 4 To check the solution, substitute it into each original equation. Then simplify.

Substitute -5 for x and for y into each original equation.

Then simplify.

In each case, the result will be a true statement.

The details of the check are left to you.

 

Note:

If we graphed the system, the lines would intersect at the point

 
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