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Solving Linear Systems of Equations by Substitution

Graphing is not always the best way to find the solution of a system of equations. It may be difficult to read the coordinates of the point of intersection. This is especially true when the coordinates are not integers.

Instead we can use algebraic methods to solve the system. One algebraic method for finding the solution of a linear system is the substitution method.

 

Procedure — To Solve a Linear System By Substitution

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

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

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

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

 

Example

Use substitution to find the solution of this system.

2x + y = 4 First equation

3x + y = 7 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 first equation for y.

Subtract 2x from both sides.

The equation y = -2x + 4 means that y and the expression -2x + 4 are equivalent.

 

2x + y

y

 

= 4

= -2x + 4

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

 

Substitute -2x + 4 for y in the second equation.

Combine like terms.

Subtract 4 from both sides.

Now we know x = 3.

Next, we will find y.

3x + y

3x + (-2x + 4)

x + 4

x

= 7

= 7

= 7

=3

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.

Substitute 3 for x.

Simplify.

The solution of the system is x = 3 and y = -2.

The solution may also be written as (3, -2).

y

y

y

= -2x + 4

= -2(3) + 4

= -2

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

Substitute x = 3 and y = -2 into both of the original equations:

  First equation   Second equation
 

Is

Is

2x

2(3)

6

+

+

-

y

(-2)

2

4

= 4

= 4 ?

= 4 ?

= 4 ? Yes

 

Is

Is

Is

3x

3(3)

9

+

+

-

y

(-2)

2

7

= 7

= 7 ?

= 7 ?

= 7 ? Yes

 

Since (3, -2) satisfies both equations, it is the solution of the system.

Note:

If we graphed the system, the lines would intersect at the point (3, -2).

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