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NES Mathematics - WEST (304): Practice & Study Guide56 chapters | 430 lessons | 24 flashcard sets

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will take a look at how to solve non-linear systems of equations in two variables by exploring two solving techniques for these types of systems - graphing and substitution.

Suppose you decide to take up gardening, but to do so, you need a garden! You decide to make one in your backyard. You want it to be rectangular, and you want it to have an area of 336 square feet.

You head to the store to buy some fencing, and a store worker says that they only have 80 feet of the fencing you want. You buy the 80 feet of fencing and head home to get started on this garden.

You have some criteria for your garden. You know that you want the area to be 336 square feet, and because you only have 80 feet worth of fencing, the distance around the rectangular garden, or the perimeter, must be 80 feet.

You need to know what the length and width of the garden need to be to fit your criteria. Well, I've got good news! We're going to see how to do just this!

You know the area and the perimeter, so if we let the length be *l*, and the width be *w*, the following equations must be true.

2*l* + 2*w* = 80

*lw* = 336

In mathematics, a **system of equations** is a set of two or more equations with the same variables. When one or more of the equations in a system are non-linear, then it is called a **non-linear system of equations**. Notice in our system, there is a non-linear equation *lw* = 336, and there are two unknowns, so our system is called a **non-linear system of equations in two variables**.

Okay, now that we have the vocabulary out of the way, let's take a look at how to solve these types of systems so you can get started on your garden!

One way to solve a non-linear system of equations in two variables is by graphing. One word of caution, however! When solving by graphing, it's best to use a graphing calculator. Doing it by hand leaves a lot of room for error since solving in this way involves locating an exact point.

The steps we use to solve by graphing are as follows.

- Graph both equations on the same graph.
- Find the intersection point of the two graphs.
- The intersection points are your solutions.

Okay! Three easy steps - we can handle this! Let's solve our system by graphing! The first step is to graph both equations on the same graph.

Next, we locate the intersection points of the two graphs.

We see the two intersection points are (12, 28) and (28, 12). The third step is the easiest! It just states that the intersection points are our solutions. In our case, we have that the length and width of the garden must be 12 feet and 28 feet to meet your criteria. Don't let the fact that there are two solutions trip you up. In this case, it just tells us that the garden will meet your criteria if it has length 12 feet and width 28 feet or length 28 feet and width 12 feet.

Awesome! You know what you need your length and width to be, but then you get to thinking, what if you didn't have a graphing calculator handy? Could you solve this system by hand another way? The answer is yes!

Another way to solve non-linear systems of equations in two variables is to do so algebraically using substitution. To use substitution to solve a system in *x* and *y*, we follow these steps.

- Solve for
*y*in terms of*x*in one of the equations. - Plug the expression you found for
*y*into the other equation, and solve for*x*. - Plug the values you found for
*x*into either of the original equations to find a corresponding value of*y*for each value of*x*. These are your solutions.

Alright! Let's look at the gardening example again. One step at a time, let's go ahead and solve the system using substitution. First, we solve for one of the variables in terms of the other variable. It doesn't matter which variable you solve for. In the end, you will get the same solutions, so let's just say that we'll solve for *l* in terms of *w* in the equation 2*l* + 2*w* = 80

We see that *l* = 40 - *w*. The next step is to plug this expression in for *l* in the other equation, *lw* = 336, and solve for *w*.

We get that *w* = 12 or *w* = 28. The last step is to plug these values into either of the original equations and solve for *l*. Let's plug them into the equation *lw* = 336.

We see that when *w* = 12, *l* = 28 and when *w* = 28, *l* = 12. In other words, your garden will meet your criteria if the width is 12 feet and the length is 28 feet or if the width is 28 feet and the length is 12 feet. This is exactly what we discovered when we solved by graphing, so we know this is correct!

A **non-linear system of equations in two variables** is a system of equations that has at least one non-linear equation and has two unknowns. We can solve these types of systems using graphing or substitution. The steps we follow in both cases are displayed in the image.

As we've seen, with your garden example, both procedures aren't too hard when we take them one step at a time! Speaking of which, thanks to mathematics, you have your garden dimensions, so go on and get started on your garden!

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NES Mathematics - WEST (304): Practice & Study Guide56 chapters | 430 lessons | 24 flashcard sets

- How to Solve a System of Linear Equations in Two Variables 4:43
- How to Solve a Linear System in Three Variables With a Solution 5:01
- How to Solve a Linear System in Three Variables With No or Infinite Solutions 6:04
- How to Write an Augmented Matrix for a Linear System 4:21
- Solving Systems of Linear Equations in Two Variables Using Determinants 4:54
- Solving Systems of Linear Equations in Three Variables Using Determinants 7:41
- Solving Systems of Nonlinear Equations in Two Variables
- Go to WEST Math: Systems of Equations

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