A system of equations includes two or more equations that are graphed on the same coordinate plane. One or more of these equations may be a quadratic equation, which takes the form of *y *= *ax*^{2} + *bx *+ *c*. We will deal with systems that include a linear and a quadratic equation, and those that contain two quadratic equations.

A system of equations may contain a linear equation and a quadratic equation. The solution to a system of equations is where the graphs of the equations intersect. A system of a linear and quadratic equation may have no solutions, one solution, or two solutions.

*y* = 2*x* + 7

*y* = *x*^{2} + 2*x* + 3

To solve a system of a linear and a quadratic equation, we simply set them equal to each other and solve for x.

The system of equations has two solutions: (2, 11) and (-2, 3).

**Graph the Solution**

Below is the graphical solution to the previous example:

- Notice that the two points of intersection are the points we solved for algebraically.
- It is also possible to solve a system of equations by graphing each equation, and finding the points where they intersect, but this is more time consuming than solving for the points algebraically.

- A system of equations that contains two quadratics can be solved in the same way as a system with a linear and a quadratic equation.
- A system with two quadratic equations will either have no solution, one solution, two solutions, or infinitely many solutions.

**Example Problem**

Solve the system of quadratic equations:

*y*=*x*^{2}+ 3*x*+ 2*y*=*x*^{2}- 2*x*- 2

The system of equations has one solution: (−2, 0), which looks like:

We can use these methods to solve not only the solutions of systems of quadratics, but for solutions of systems of any type of equation, including the various types of conic sections. We can find the intersections, or solutions, of any two equations by setting them equal to one another or using substitution.

**For Example**

Find the solution to the system of equations below:

- 2
*x*^{2}+ 3y^{2}+ 3*x*- 12*y*- 42 = 0 *x*+ 3*y*= 0

The first equation is an equation of an ellipse, and the second is a linear equation. We are trying to find the points in which the line intersects the ellipse.

Since these equations each contain two variables, the easiest approach is to substitute the linear equation into the ellipse equation by solving for a variable.

**Which looks like this when graphed:**

**Answers to Practice Problems**

- 2
- -4
- (9, 2), (−3, −2)
- 0, no solutions
- 0

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