7. Systems of Equations & Matrices

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

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 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 + 3y = 2 3x + 9y = 6

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

$3x^2+5y^2=17$

$2x^2-3y^2=5$

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

$x^2+y^2=102$

$x^2-y^2=17$

In Exercises 5–18, solve each system by the substitution method. 2x + 5y = - 4 3x - y = 11

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations.

2x² = 3y + 23

y = 2x - 5

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

5x + 7y = 6

10x - 3y = 46

Solve each system by substitution.

8x - 10y = -22

3x + y = 6

The perimeter of a table tennis top is 28 feet. The difference between 4 times the length and 3 times the width is 21 feet. Find the dimensions.

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Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations.

y = 3x²

x² + y² = 10

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

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,____).

x + y = 7

x² + y² = 25

In Exercises 19–30, solve each system by the addition method. 3x - 4y = 11 2x + 3y = - 4

Determine the system of equations illustrated in each graph. Write equations in standard form.