7. Systems of Equations and Inequalities

In Exercises 1–4, determine whether the given ordered pair is a solution of the system. (2, 5) 2x + 3y = 17 x + 4y = 16

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. 1/6x + 1/3y = 8 1/4x + 1/2y = 12

A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of a mixture that is 7% silver nitrate. How many milliliters of each of the solutions must be used?

In Exercises 5–18, solve each system by the substitution method. x + 3y = 8 y = 2x - 9

Solve each system by substitution. See Example 1. x - 5y = 8 x = 6y

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

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

Solve each problem using a system of equations. A company sells recordable CDs for $0.80 each and play-only CDs for $0.60 each. The company receives $76.00 for an order of 100 CDs. However, the customer neglected to specify how many of each type to send. Determine the number of each type of CD that should be sent.

In Exercises 5–18, solve each system by the substitution method. y = (1/3)x + 2/3 y = (5/7)x - 2

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. 4x + y = -23 x - 2y = -17

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

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

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. y = 3x - 5 21x - 35 = 7y

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

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

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.

In Exercises 47–48, solve each system by the method of your choice. (x - y)/3 = (x + y)/2 - 1/2 (x + 2)/2 - 4 = (y + 4)/3

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?

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.

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. 2x + 6y = 6 5x + 9y = 9

In Exercises 1–4, determine whether the given ordered pair is a solution of the system. (- 3, 5) 9x + 7y = 8 8x - 9y = - 69

In Exercises 1–5, 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.

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

Solve each system by substitution. See Example 1. 4x + 3y = -13 -x + y = 5

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.

In Exercises 5–18, solve each system by the substitution method. 5x + 2y = 0 x - 3y = 0

In Exercises 5–18, solve each system by the substitution method. 2x - 3y = 8 - 2x 3x + 4y = x + 3y + 14

Solve each system by substitution. See Example 1. 3y = 5x + 6 x + y = 2

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

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

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. (2x-1)/3 + (y+2)/4 = 4 (x+3)/2 - (x-y)/2 = 3

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 = 9-2y x + 2y = 13

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. 3x - 2y = − 5 4x + y = 8

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. 2x = 3y + 4 4x = 3 - 5y

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

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

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?

Use a system of equations to solve each problem. See Example 8. Find an equation of the parabola y = ax^2 + bx + c that passes through the points (2, 3), (-1, 0), and (-2, 2).

In Exercises 1–4, determine whether the given ordered pair is a solution of the system. (2, 3) x + 3y = 11 x - 5y = - 13

In Exercises 1–5, 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.

A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of a mixture that is 7% silver nitrate. How many milliliters of each of the solutions must be used?

In Exercises 5–18, solve each system by the substitution method. x = 4y - 2 x = 6y + 8

Solve each system by substitution. See Example 1. -2x = 6y + 18 -29 = 5y - 3x

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

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. 6x + 7y + 2 = 0 7x - 6y - 26 = 0

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. x/2+ y/3 = 4 3x/2+3y/2 = 15

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

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. See Examples 3 and 4. 5x - 5y - 3 = 0 x - y - 12 = 0

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

In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 5ax + 4y = 17 ax + 7y = 22

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).

In Exercises 1–5, 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.

In Exercises 5–18, solve each system by the substitution method. 5x + 2y = 0 x - 3y = 0

Solve each system by substitution. See Example 1. 7x - y = -10 3y - x = 10

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

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. 5x + 7y = 6 10x - 3y = 46

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. See Examples 3 and 4. 9x - 5y = 1 -18x + 10y = 1

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

In Exercises 47–48, solve each system by the method of your choice. (x - y)/3 = (x + y)/2 - 1/2 (x + 2)/2 - 4 = (y + 4)/3

Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.) 2/x + 3/y = 18 4/x - 5/y = -8

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions? x - 2y = 3 -2x + 4y = k

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?

In Exercises 1–5, 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.

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. 6x + 10y = -11 9x + 6y = -3

In Exercises 5–18, solve each system by the substitution method. x = 4y - 2 x = 6y + 8

Solve each system by substitution. See Example 1. 8x - 10y = -22 3x + y = 6

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

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. 2x - 3y = -7 5x + 4y = 17

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

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

In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 5ax + 4y = 17 ax + 7y = 22

Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.) 2/x + 1/y = 3/2 3/x - 1/y = 1

Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for $305, while 2 goats and 9 sheep sell for $285. Find the cost of a single goat and of a single sheep.