7. Systems of Equations & Matrices

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).

In Exercises 1–18, solve each system by the substitution method. $\(\begin{cases}\)xy = 6 \\2x - y = 1\(\end{cases}\)$

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose sum is 17 and whose product is 42.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.

y = x² - 2x + 1

x - 3y = -1

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

$2x+y=1$

$3x-y=4$

In Exercises 19–30, solve each system by the addition method. 4x + 3y = 15 2x - 5y = 1

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

$\{\begin{array}{c}y=4x+1\\ 3x+2y=13\end{array}$

In Exercises 29–42, solve each system by the method of your choice. $\(\begin{cases}\)x^2 + 4y^2 = 20 \\(\x\) + 2y = 6\(\end{cases}\)$

Solve each problem using a system of equations in two variables. See Example 6. The longest side of a right triangle is 13 m in length. One of the other sides is 7 m longer than the shortest side. Find the lengths of the two shorter sides of the triangle.

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose squares have a sum of 100 and a difference of 28.

In Exercises 47–52, solve each system by the method of your choice. $\(\begin{cases}\]\frac{3}{x^2}\) + \(\frac{1}{y^2}\) = 7 \\\(\frac{5}{x^2}\) - \(\frac{2}{y^2}\) = -3\(\end{cases}\)$

In Exercises 47–48, solve each system by the method of your choice. (x + 2)/2 - (y + 4)/3 = 3 (x + y)/5 = (x - y)/2 - 5/2

Solve each system by substitution.

x - 5y = 8

x = 6y

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

$y=5x-17$

$15x-3y=51$

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