7. Systems of Equations & Matrices

Use the elimination method to solve the following system of linear equations.

$10x-4y=5$

$5x-4y=1$

In Exercises 19–28, solve each system by the addition method. $\(\begin{cases}\)x^2 + y^2 = 25 \\(x - 8)^2 + y^2 = 41\(\end{cases}\)$

In Exercises 19–30, solve each system by the addition method. 2x + 3y = 6 2x - 3y = 6

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. If we want to solve the following nonlinear system by substitution and we decide to solve equation (2) for y, what will be the resulting equation when the substitution is made into equation (1)?

x² + y = 2 (1)

x - y = 0 (2)

Solve each system by the method of your choice.

$\{\begin{array}{c}5y=x^2-1\\ x-y=1\end{array}$

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 equations. Solve the system and find the numbers. The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

$x^2+y^2=5$

$-3x+4y=2$

Solve the following system of equations. Classify it as CONSISTENT (INDEPENDENT or DEPENDENT) or INCONSISTENT.

$2x+8y=7$

$x+4y=19$

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3).

2x + y = 1

x² + y² = 10

In Exercises 19–30, solve each system by the addition method. x + 2y = 2 - 4x + 3y = 25

Solve the systems in Exercises 79–80.

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

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

In Exercises 1–18, solve each system by the substitution method. $\(\begin{cases}\)y^2 = x^2 - 9 \\2y = x - 3\(\end{cases}\)$

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.

$\frac{1}{6x}+\frac{1}{3y}=8$

$\frac{1}{4x}+\frac{1}{2y}=12$

Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary.

9x - 5y = 1

-18x + 10y = 1