A system of equations is a set of 2 or more equations, where the variables represent the same unknown values. For example, suppose that two different kinds of bamboo are planted at the same time. Plant A starts at 6 ft tall and grows at a constant rate of $\frac14$ foot each day. Plant B starts at 3 ft tall and grows at a constant rate of $\frac12$ foot each day. Because Plant B grows faster than Plant A, it will eventually be taller, but when?

Solving a system of equations means to find the values of $x$ and $y$ that make both equations true at the same time. One way we have seen to find the solution to a system of equations is to graph both lines and find the intersection point. The intersection point represents the pair of $x$ and $y$ values that makes both equations true.

Here is a graph for the bamboo example:

<p>Graph of two lines, origin O, with grid. Horizontal axis, time in days, scale 0 to 13, by 1’s. Vertical axis, height in feet, scale 0 to 12, by 1’s. A line, labeled Plant A, crosses the y axis at 6. A line, labeled Plant B, crosses the y axis at 3. The lines intersect at the point 12 comma 9.</p>  

The solution to this system of equations is $(12,9)$, which means that both bamboo plants will be 9 feet tall after 12 days.

We have seen systems of equations that have no solutions, one solution, and infinitely many solutions.

• When the lines do not intersect, there is no solution. (Lines that do not intersect are parallel.)
• When the lines intersect once, there is one solution.
• When the lines are right on top of each other, there are infinitely many solutions.