7. Systems of Equations & Matrices

Solve each problem. Alcohol Mixture Barak wishes to strengthen a mixture that is 10% alcohol to one that is 30% alcohol. How much pure alcohol should he add to 12 L of the 10% mixture?

In Exercises 1–18, solve each system by the substitution method. $\(\begin{cases}\)x + y = 2 \\(\y\) = x^2 - 4x + 4\(\end{cases}\)$

In Exercises 25–35, solve each system by the method of your choice. This is a piecewise function, refer to textbook problem.

In Exercises 1–18, solve each system by the substitution method.$\(\begin{cases}\)x^2 + y^2 = 25 \\(\x\) - y = 1\(\end{cases}\)$

The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.

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

2xy + 1 = 0

x + 16y = 2

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 47–52, solve each system by the method of your choice. $\(\begin{cases}\)-4x + y = 12 \\(\y\) = x^3 + 3x^2\(\end{cases}\)$

Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x² + y² = 25? (Hint: To find out, solve the system formed by these two equations.)

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. $\(\begin{cases}\)3x^2 + 4y^2 = 16 \\2x^2 - 3y^2 = 5\(\end{cases}\)$

In Exercises 5–18, solve each system by the substitution method. $\(\begin{cases}\)x = 4y - 2 \\(\x\) = 6y + 8\(\end{cases}\)$

Solve each system by elimination. In systems with fractions, first clear denominators.

4x + y = -23

x - 2y = -17

Solve the systems in Exercises 79–80.

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

In Exercises 31–42, 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. x/4 - y/4 = −1 x + 4y = -9