Solving a System with Infinitely Many Solutions

Example

Solve the system: 

$8x-2y=-4$

$-4x+y=2$

Solution

Divide both sides of the first equation by 2, and add the result to the second equation.

$4x-y=-2$

$-4x+y=2$

_______________

$0=0$

The last equation is clearly true.

Since the result, 0 = 0, is a true statement, which indicates that the equations of the original system are equivalent. Any ordered pair (x, y) that satisfies either equation will satisfy the system. Solve for y in the second equation.

$-4x+y=2$

$y=4x+2$

The solutions of the system can be written in the form of a set of ordered pairs (x, 4x + 2), for any real number x. Some ordered pairs in the solution set are (0, 4(0)+2), or (0, 2), and (1, 4(1)+2) or (1, 6), as well as (3, 14), and (-2, -6).

As shown here, the equations of the original system are dependent and lead to the same straight-line graph. The solution set can be written {(x, 4x + 2)}.

Note

In the algebraic solution for the example above, we wrote the solution set with the variable x arbitrary. We could write the solution set with y arbitrary.

$\lbrace\left(\frac{y-2}{4}\right),y\rbrace$

By selecting values for y and solving for x in this ordered pair, we can find individual solutions. Verify again that (0, 2) is a solution by letting y = 2 and solving for x to obtain $\frac{2-2}{4}=0$.