Solving Systems of Equations

Student Summary

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

{y=[some stuff]y=[some other stuff]\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)(x,y) that makes both equations true. In particular, we know that the value for yy will be the same in both equations. That means that

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

For example, look at this system of equations:

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

Since the yy value of the solution is the same in both equations, then we know that:

2x+6=-3x−42x + 6 = \text-3x -4

We can solve this equation for xx:

\begin{aligned} 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 xx, but we need the corresponding value for yy.

Since both equations have the same yy value, we can use either equation to find the yy-value: 2(-2)+62(\text-2) + 6 or y=-3(-2)−4y = \text-3(\text-2) -4.

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

&lt;p&gt;Graph of two lines. &lt;/p&gt;<br>  
<p>Graph of two lines line, origin O, with grid. Horizontal axis, x, scale negative 4 to 1, by 1s. Vertical axis, y, scale negative 1 to 4, by 1’s. The lines intersect at the point negative 2 comma 2. </p>  

In general, a system of linear equations can have:

  • No solutions. In this case, the lines that correspond to each equation never intersect. They have the same slope and different yy-intercepts.
  • Exactly one solution. The lines that correspond to each equation intersect in exactly one point. They have different slopes.
  • An infinite number of solutions. The graphs of the two equations are the same line! They have the same slope and the same yy-intercept.

Visual / Anchor Chart

Standards

Addressing
8.EE.8.a

Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs. Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients.

8.EE.8

Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs. Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients.