11.5 Solve Systems of Nonlinear Equations

Learning objectives

Learning objectives

• Solve a system of nonlinear equations using graphing

• Solve a system of nonlinear equations using substitution

• Solve a system of nonlinear equations using elimination

• Use a system of nonlinear equations to solve applications

Solve a System of Nonlinear Equations Using Graphing

We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A system of nonlinear equations is a system where at least one of the equations is not linear.

For example each of the following systems is a system of nonlinear equations.

Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.

When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.

To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.

Example 11.33

Solution

To identify the graph of each equation, keep in mind the characteristics of the ${x}^{2}$ and ${y}^{2}$ terms of each conic.

Example 11.34

Solution

Solve a System of Nonlinear Equations Using Substitution

The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.

The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.

Example 11.35

Solution

So far, each system of nonlinear equations has had at least one solution. The next example will show another option.

Example 11.36

Solution

Solve a System of Nonlinear Equations Using Elimination

When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.

The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.

Example 11.37

Solution

There are also four options when we consider a circle and a hyperbola.

Example 11.38

Solution

Use a System of Nonlinear Equations to Solve Applications

Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.

Example 11.39

Solution

Example 11.40

Solution

Section 11.5 Exercises

Section 11.5 Exercises

Solve a System of Nonlinear Equations Using Graphing

In the following exercises, solve the system of equations by using graphing.

Solve a System of Nonlinear Equations Using Substitution

In the following exercises, solve the system of equations by using substitution.

Solve a System of Nonlinear Equations Using Elimination

In the following exercises, solve the system of equations by using elimination.

Use a System of Nonlinear Equations to Solve Applications

In the following exercises, solve the problem using a system of equations.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

1. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} x+y=4 \\ y={x}^{2}+2 \\ \end{array}.$

2. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} x-y=-1 \\ y=-{x}^{2}+3 \\ \end{array}.$

3. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} x=-6 \\ {(x+3)}^{2}+{(y-1)}^{2}=9 \\ \end{array}.$

1. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} y=4 \\ {(x-2)}^{2}+{(y+3)}^{2}=4 \\ \end{array}.$

2. Solve the system by using substitution: $\mathrm{\{}\begin{array}{l} {x}^{2}+9{y}^{2}=9 \\ y=\frac{1}{3}x-3 \\ \end{array}.$

3. Solve the system by using substitution: $\mathrm{\{}\begin{array}{l} 4{x}^{2}+{y}^{2}=4 \\ y=x+2 \\ \end{array}.$

1. Solve the system by using substitution: $\mathrm{\{}\begin{array}{l} {x}^{2}-y=0 \\ y=2x-3 \\ \end{array}.$

2. Solve the system by using substitution: $\mathrm{\{}\begin{array}{l} {y}^{2}-x=0 \\ y=3x-2 \\ \end{array}.$

3. Solve the system by elimination: $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=9 \\ {x}^{2}-y=9 \\ \end{array}.$

1. Solve the system by elimination: $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=1 \\ -x+{y}^{2}=1 \\ \end{array}.$

2. Solve the system by elimination: $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=25 \\ {y}^{2}-{x}^{2}=7 \\ \end{array}.$

3. Solve the system by elimination: $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=4 \\ {x}^{2}-{y}^{2}=4 \\ \end{array}.$

1. The difference of the squares of two numbers is $-20.$ The sum of the numbers is 10. Find the numbers.

2. The difference of the squares of two numbers is 35. The sum of the numbers is $-1.$ Find the numbers.

3. Edgar purchased a small 20” TV for his garage. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 192 square inches. What are the length and width of the TV screen?

1. The Harper family purchased a small microwave for their family room. The diagonal of the door measures 15 inches. The door also has an area of 108 square inches. What are the length and width of the microwave door?

2. $\mathrm{\{}\begin{array}{l} y=6x-4 \\ y=2{x}^{2} \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} x-y=-2 \\ x={y}^{2} \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y=x-1 \\ y={x}^{2}+1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} y=-4 \\ {x}^{2}+{y}^{2}=16 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y=-1 \\ {(x-2)}^{2}+{(y-4)}^{2}=25 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y=-\frac{1}{2}x+2 \\ y=\sqrt[]{x}-2 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 9{x}^{2}+{y}^{2}=9 \\ y=3x+3 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 9{x}^{2}+4{y}^{2}=36 \\ x=2 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=169 \\ x=12 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 2{y}^{2}-x=0 \\ y=x+1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y={x}^{2}-4 \\ y=x-4 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=25 \\ 2x+y=10 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=16 \\ {x}^{2}-y=4 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=4 \\ {x}^{2}-y=2 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=4 \\ {y}^{2}-x=2 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=20 \\ {x}^{2}-{y}^{2}=-12 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} {x}^{2}+{y}^{2}=16 \\ {x}^{2}-{y}^{2}=16 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} {x}^{2}-{y}^{2}=3 \\ 2{x}^{2}+{y}^{2}=6 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} {x}^{2}-{y}^{2}=-5 \\ 3{x}^{2}+2{y}^{2}=30 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 2{x}^{2}+{y}^{2}=11 \\ {x}^{2}+3{y}^{2}=28 \\ \end{array}$

1. The sum of two numbers is 11 and the product is $-42.$ Find the numbers.

2. The sum of the squares of two numbers is 113. The difference of the numbers is 1. Find the numbers.

3. The difference of the squares of two numbers is 20. The difference of the square of the first number and twice the square of the second number is 4. Find the numbers.

1. The perimeter of a rectangle is 52 cm and its area is 165 ${\mathrm{cm}}^{2}.$ Find the length and width of the rectangle.

2. Jules purchased a microwave for his kitchen. The diagonal of the front of the microwave measures 26 inches. The front also has an area of 240 square inches. What are the length and width of the microwave?

3. Donnette found a widescreen TV at a garage sale, but isn’t sure if it will fit her entertainment center. The TV is 50”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1200 square inches. Her entertainment center has an insert for the TV with a length of 38 inches and width of 27 inches. What are the length and width of the TV screen and will it fit Donnette’s entertainment center?

1. Explain in your own words how to solve a system of equations using substitution.

2. A circle and a parabola can intersect in ways that would result in 0, 1, 2, 3, or 4 solutions. Draw a sketch of each of the possibilities.