
Hi, and welcome to this video on solving systems of equations with a linear and a quadratic equation!
In this video, we will take a look at two different ways to approach these problems: graphically and algebraically.
Let’s get started!
Linear Equations vs. Quadratic Equations
Before we get into how to solve these systems, let’s review the differences between linear and quadratic equations.
Linear equations are equations that graph as a line. They are equations that can be written as
Quadratic equations are equations of the form
Solving Systems of Equations
When we solve systems of equations, we are taking two or more equations and finding the point, or points, where they intersect. When we have a linear equation and a quadratic equation, we will have zero, one, or two points of intersection.
Example #1
The first solving method we are going to look at is solving graphically. To do this, we simply graph our equations, either with a calculator or by hand, and see where the graphs intersect.
Let’s take a look at the equations:
If we graph these two equations next to each other, we get something that looks like this:
We can see from our graph that the two equations intersect at the points negative 4,0) and (7,22). (-4,0) and (7,22).
But what if we aren’t able to graph our equations? Well, that’s where algebra comes in. The other way to solve systems like this is very similar to the substitution method of solving regular systems of linear equations.
To find at what points our equations are equal to each other, we can substitute our
When we write it out, that will look like this:
In order to solve a quadratic equation for
That gives us:
We can then factor this equation to get:
This tells us that our zeroes for this quadratic equation are -4 and 7. These are our values for
To find our
So our points of intersection for this graph are
If we wanted one more way to check our answers, we can plug them back into our original equations and make sure our answers match up.
Example #2
Let’s try one more example. Solve this system graphically and then algebraically:
In order to graph our linear equation, we want to first rearrange it so that y is by itself on one side. We can do this by adding y and subtracting 1 from both sides.
When we do this, it gives us:
Now we are able to graph both equations, and our graph will look like this:
We can see that our graphs intersect at the points
Now let’s check and see if we get the same answers using algebra.
If we use our rearranged linear equation, we are able to set our two equations equal to each other and solve.
We can subtract 12x and add 1 to both sides to get all of our terms on one side and zero on the other.
Now we want to factor our equation to get our x-values.
If we set each equation individually to equal zero, we get:
On this side, to solve for
On this side, we just simply add 9 to both sides.
These are our
So, we’re going to take our original equation,
Our two points of intersection are
There’s one important thing to point out before we go. In both of these examples, our equations factored nicely to give us our
I hope this video on solving systems with a linear and a quadratic function was helpful. Thanks for watching and happy studying!
System of Equation Practice Questions
The equations
Which graph could be used to identify the solutions to the following system of equations?




Two bus routes in Sunnydale can be graphed using the equations
Which ordered pair is a possible solution to the system of equations shown below? Use a graph to help you solve.
Stephanie is a city planner who is analyzing a map of street intersections to see where stoplights need to be installed. Specifically, she is looking at two streets that can be graphed using the equations