11.5 Solve Systems of Nonlinear Equations

Solve the system by graphing: $\begin{cases} x - 3 y = −3 \\ x + y = 5 \end{cases} .$
If you missed this problem, review the reference.

Solve the system by substitution: $\begin{cases} x - 4 y = −4 \\ - 3 x + 4 y = 0 \end{cases} .$
If you missed this problem, review the reference.

Solve the system by elimination: $\begin{cases} 3 x - 4 y = −9 \\ 5 x + 3 y = 14 \end{cases} .$
If you missed this problem, review the reference.

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

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

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

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

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

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

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

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

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

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

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?

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?

Solve the system of equations by using substitution.

$\begin{cases} 9 x^{2} + y^{2} = 9 \\ y = 3 x + 3 \end{cases}$

Solve the system of equations by using substitution.

$\begin{cases} 9 x^{2} + 4 y^{2} = 36 \\ x = 2 \end{cases}$

Solve the system of equations by using substitution.

$\begin{cases} x^{2} + y^{2} = 169 \\ x = 12 \end{cases}$

Solve the system of equations by using substitution.

$\begin{cases} 2 y^{2} - x = 0 \\ y = x + 1 \end{cases}$

Solve the system of equations by using substitution.

$\begin{cases} y = x^{2} - 4 \\ y = x - 4 \end{cases}$

Solve the system of equations by using substitution.

$\begin{cases} x^{2} + y^{2} = 25 \\ 2 x + y = 10 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} x^{2} + y^{2} = 16 \\ x^{2} - y = 4 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} x^{2} + y^{2} = 4 \\ x^{2} - y = 2 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} x^{2} + y^{2} = 4 \\ y^{2} - x = 2 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} x^{2} + y^{2} = 20 \\ x^{2} - y^{2} = −12 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} x^{2} + y^{2} = 16 \\ x^{2} - y^{2} = 16 \end{cases}$

Solve the system of equations by using elimination.

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

Solve the system of equations by using elimination.

$\begin{cases} x^{2} - y^{2} = −5 \\ 3 x^{2} + 2 y^{2} = 30 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} 2 x^{2} + y^{2} = 11 \\ x^{2} + 3 y^{2} = 28 \end{cases}$

Solve the problem using a system of equations.

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

Solve the problem using a system of equations.

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

Solve the problem using a system of equations.

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.

Solve the problem using a system of equations.

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

Solve the problem using a system of equations.

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?

Solve the problem using a system of equations.

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 lengt…

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

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.

Find the distance between the points. Round to the nearest tenth if needed.

$\left(−2 , 5\right)$ and $\left(1 , 5\right)$

Find the distance between the points. Round to the nearest tenth if needed.

$\left(1 , −4\right)$ and $\left(5 , −5\right)$

Find the midpoint of the line segment whose endpoints are given.

$\left(3 , 7\right)$ and $\left(5 , 1\right)$

Find the midpoint of the line segment whose endpoints are given.

$\left(−3 , 2\right)$ and $\left(6 , −9\right)$

Write the standard form of the equation of the circle with the given information.

radius is $\sqrt{7}$ and center is $\left(0 , 0\right)$

Write the standard form of the equation of the circle with the given information.

radius is 7 and center is $\left(−2 , −5\right)$

Write the standard form of the equation of the circle with the given information.

center is $\left(2 , 2\right)$ and a point on the circle is $\left(4 , 4\right)$

Identify the type of graph.

ⓐ $16 y^{2} - 9 x^{2} - 36 x - 96 y - 36 = 0$
ⓑ $x^{2} + y^{2} - 4 x + 10 y - 7 = 0$
ⓒ $y = x^{2} - 2 x + 3$
ⓓ $25 x^{2} + 9 y^{2} = 225$

Solve the system of equations by using substitution.

$\begin{cases} y = x^{2} + 3 \\ y = −2 x + 2 \end{cases}$

Solve the system of equations by using substitution.

$\begin{cases} 9 x^{2} + 4 y^{2} = 36 \\ y - x = 5 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} x^{2} + y^{2} = 16 \\ x^{2} - 2 y - 1 = 0 \end{cases}$

Solve the system of equations by using elimination.

$\begin{cases} 4 x^{2} + 9 y^{2} = 36 \\ 3 y^{2} - 4 x = 12 \end{cases}$

Solve the problem using a system of equations.

The sum of the squares of two numbers is 25. The difference of the numbers is 1. Find the numbers.

Solve the problem using a system of equations.

The perimeter of a rectangle is 58 meters and its area is 210 square meters. Find the length and width of the rectangle.

Find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.

$\left(−4 , −3\right)$ and $\left(−10 , −11\right)$

Write the standard form of the equation of the circle with the given information.

radius is 11 and center is $\left(0 , 0\right)$

Write the standard form of the equation of the circle with the given information.

center is $\left(−2 , 3\right)$ and a point on the circle is $\left(2 , −3\right)$

ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation.

Solve the nonlinear system of equations using elimination:
$\begin{cases} x^{2} + 9 y^{2} = 9 \\ 2 x^{2} - 9 y^{2} = 18 \end{cases} .$

ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation.

For her birthday, Olive’s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is 55”. The size of a TV is measured on the diagonal of the screen and a widescreen ha…