7. Systems of Equations & Matrices

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 50% alcohol solution must be mixed with 80 ounces of a 20% alcohol solution to make a 40% alcohol solution?

Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

Use a system of equations to solve each problem. Find an equation of the parabola y = ax² + bx + c that passes through the points (2, 3), (-1, 0), and (-2, 2).

Exercises 86–88 will help you prepare for the material covered in the first section of the next chapter. a. Does (4, −1) satisfy x + 2y = 2? b. Does (4, -1) satisfy x- 2y= 6?

Solve each problem. See Examples 5 and 9. The sum of two numbers is 47, and the difference between the numbers is 1. Find the numbers.

Solve the system

Solve the systems in Exercises 79–80.

$\{\begin{array}{c}\text{log }{x}^2=y+3\\ \text{log }{x}^{}=y-1\end{array}$

Solve the systems in Exercises 79–80.

$\{\begin{array}{c}{\log }_y\text{ x=3}\\ {\log }_y\text{ (4x)=5}\end{array}$

Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is 3. Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}x^2 + y^2 + 3y = 22 \\2x + y = -1\end{cases}$

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}y = (x + 3)^2 \\x + 2y = -2\end{cases}$

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}x^2 + (y - 2)^2 = 4 \\x^2 - 2y = 0\end{cases}$