MA001: College Algebra

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as $0=0$.

Example 9

Finding a Solution to a Dependent System of Linear Equations

Find a solution to the system of equations using the addition method.

$x+3y=2$

$3x+9y=6$

Solution
With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let's focus on eliminating $x$. If we multiply both sides of the first equation by −3, then we will be able to eliminate the x -variable.

$x+3y=2$

$(−3)(x+3y)=(−3)(2)$

$−3x−9y=−6$

Now add the equations.

$−3x−9y = −6$

$+3x+9y=6$

$0=0$

We can see that there will be an infinite number of solutions that satisfy both equations.

Analysis

If we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Let's look at what happens when we convert the system to slope-intercept form.

$x+3y=2$

$3y=−x+2$

$y=−\frac{1}{3}x+\frac{2}{3}$

$3x+9y=6$

$9y=−3x+6$

$y=−\frac{3}{9}x+\frac{6}{9}$

$y=−\frac{1}{3}x+\frac{2}{3}$

See Figure 9. Notice the results are the same. The general solution to the system is $(x, −\frac{1}{3}x+\frac{2}{3})$.

Figure 9

Try It #7

Solve the following system of equations in two variables.

$y−2x=5$

$−3y+6x=−15$