### Solving Systems of Linear Equations by Substitution

Graphing is a useful tool for solving systems of equations, but it can sometimes be time-consuming. A quicker way to solve systems is to isolate one variable in one equation, and substitute the resulting expression for that variable in the other equation. Observe:

*Example 1*: Solve the following system, using substitution:

The easiest variable to isolate is

*y* in the first equation, because it has no coefficient:

*y* = 13 - 5*x*
In the second equation, substitute for

*y* its equivalent expression:

3*x* = 15 - 3(13 - 5*x*)
Solve the equation:

3*x* = 15 - 39 + 15*x*
3*x* = 15*x* - 24
-12*x* = - 24
*x* = 2
Now substitute this

*x*-value into the "isolation equation" to find

*y*:

*y* = 13 - 5*x* = 13 - 5(2) = 13 - 10 = 3
Thus, the solution to the system is

(2, 3). It is useful to check this solution in both equations.

**Note:** Although we chose *y* in the first equation in the previous example, isolating any variable in any equation will yield the same solution.

*Example 2*: Solve the following system, using substitution:

It is easier to work with the second equation, because there is no constant term:

5*x* = 10*y*
*x* = 2*y*
In the first equation, substitute for

*x* its equivalent expression:

2(2*y*) + 4*y* = 36
Solve the equation:

4*y* + 4*y* = 36
8*y* = 36
*y* = 4.5
Plug this

*y*-value into the isolation equation to find

*x*:

*x* = 2*y* = 2(4.5) = 9
Thus, the solution to the system is (9, 4.5).

*Example 3*: Solve the following system, using substitution:

It is easiest to isolate

*x* in the second equation, since the

*x* term already stands alone:

*x* =
*x* = 7 + 2*y*
In the first equation, substitute for

*x* its equivalent expression:

2(7 + 2*y*) - 4*y* = 12
Solve the equation:

14 + 4*y* - 4*y* = 12
14 = 12
Since

14≠12, the system of equations has no solution. It is inconsistent (and independent). The two equations describe two parallel lines.

*Example 4*: Solve the following system, using substitution:

Either equation can be used to isolate the variable. We will isolate

*y* in the second equation:

2*y* = 5*x* + 34
*y* =
*y* = *x* + 17
In the first equation, substitute for

*y* its equivalent expression:

10*x* = 4(*x* + 17) - 68
10*x* = 10*x* + 68 - 68
10*x* = 10*x*
0 = 0
Since

0 = 0 for any value of

*x*, the system of equations has infinite solutions. Every ordered pair (

*x*, *y*) which satisfies

*y* = *x* + 17 (the isolation equation) is a solution to the system. The system is dependent (and consistent). The two equations describe the same line--

*y* = *x* + 17.