Lesson 13: Solving Systems of Equations

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

$$\begin{cases} y = \text{[some stuff]}\\ y = \text{[some other stuff]} \end{cases}$$

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

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

For example, look at this system of equations:

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

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

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:

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

Or

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

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!