Systems of Equations

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 - 5x
In the second equation, substitute for y its equivalent expression:
3x = 15 - 3(13 - 5x)
Solve the equation:
3x = 15 - 39 + 15x
3x = 15x - 24
-12x = - 24
x = 2
Now substitute this x-value into the "isolation equation" to find y:
y = 13 - 5x = 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:

5x = 10y
x = 2y
In the first equation, substitute for x its equivalent expression:
2(2y) + 4y = 36
Solve the equation:
4y + 4y = 36
8y = 36
y = 4.5
Plug this y-value into the isolation equation to find x:
x = 2y = 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 + 2y
In the first equation, substitute for x its equivalent expression:
2(7 + 2y) - 4y = 12
Solve the equation:
14 + 4y - 4y = 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:

2y = 5x + 34
y =
y = x + 17
In the first equation, substitute for y its equivalent expression:
10x = 4(x + 17) - 68
10x = 10x + 68 - 68
10x = 10x
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.