Lesson 13

Solving Systems of Equations

Let’s solve systems of equations.

13.1: True or False: Two Lines

Graph of 2 lines.

Use the lines to decide whether each statement is true or false. Be prepared to explain your reasoning using the lines.

  1. A solution to \(8=\text-x+10\) is 2.
  2. A solution to \(2=2x+4\) is 8.
  3. A solution to \(\text-x+10=2x+4\) is 8.
  4. A solution to \(\text-x+10=2x+4\) is 2.
  5. There are no values of \(x\) and \(y\) that make \(y=\text-x+10\) and \(y=2x+4\) true at the same time.

13.2: Matching Graphs to Systems

Here are three systems of equations graphed on a coordinate plane:

Three graphs, with two lines each, in the x y plane.
  1. Match each figure to one of the systems of equations shown here.
    1. \(\begin{cases} y=3x+5\\ y=\text- 2x+20 \end{cases}\)

    2. \(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)

    3. \(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)

  2. Find the solution to each system and then check that your solution is reasonable on the graph.
    • Notice that the sliders set the values of the coefficient and the constant term in each equation.
    • Change the sliders to the values of the coefficient and the constant term in the next pair of equations.
    • Click on the spot where the lines intersect and a labeled point should appear.

13.3: Different Types of Systems

Your teacher will give you a page with 6 systems of equations.

  1. Graph each system of equations by typing each pair of the equations in the applet, one at a time.

  2. Describe what the graph of a system of equations looks like when it has . . .
    1. 1 solution
    2. 0 solutions
    3. infinitely many solutions

Use the applet to confirm your answer to question 2.

The graphs of the equations \(Ax + By = 15\) and \(Ax - By = 9\) intersect at \((2,1)\). Find \(A\) and \(B\). Show or explain your reasoning.


Sometimes it is easier to solve a system of equations without having to graph the equations and look for an intersection point. In general, whenever we are solving a system of equations written as

\(\displaystyle \begin{cases} y = \text{[some stuff]}\\ y = \text{[some other stuff]} \end{cases}\)

we know that we are looking for a pair of values \((x,y)\) that makes both equations true. In particular, we know that the value for \(y\) will be the same in both equations. That means that

\(\displaystyle \text{[some stuff]} = \text{[some other stuff]}\)

For example, look at this system of equations:

\(\displaystyle \begin{cases} y = 2x + 6 \\ y = \text-3x - 4 \end{cases}\)

Since the \(y\) value of the solution is the same in both equations, then we know \(\displaystyle 2x + 6 = \text-3x -4\)

We can solve this equation for \(x\):

\(\begin{align} 2x+6 &= \text-3x-4&& \\ 5x+6 &=\text-4\ &&\text{add \(3x\) to each side}\\ 5x &=\text-10\ &&\text{subtract 6 from each side}\\ x &=\text-2\ &&\text{divide each side by 5}\ \end{align}\)

But this is only half of what we are looking for: we know the value for \(x\), but we need the corresponding value for \(y\). Since both equations have the same \(y\) value, we can use either equation to find the \(y\)-value:

\(\displaystyle y = 2(\text-2) + 6\)


\(\displaystyle y = \text-3(\text-2) -4\)

In both cases, we find that \(y = 2\). So the solution to the system is \((\text-2,2)\). We can verify this by graphing both equations in the coordinate plane.

Graph of two lines. 

In general, a system of linear equations can have:

  • No solutions. In this case, the lines that correspond to each equation never intersect.
  • Exactly one solution. The lines that correspond to each equation intersect in exactly one point.
  • An infinite number of solutions. The graphs of the two equations are the same line!

Glossary Entries

  • system of equations

    A system of equations is a set of two or more equations. Each equation contains two or more variables. We want to find values for the variables that make all the equations true.

    These equations make up a system of equations:

    \(\displaystyle \begin{cases} x + y = \text-2\\x - y = 12\end{cases}\)

    The solution to this system is \(x=5\) and \(y=\text-7\) because when these values are substituted for \(x\) and \(y\), each equation is true: \(5+(\text-7)=\text-2\) and \(5-(\text-7)=12\).