7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations

7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations

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Introduction to Systems of Linear Equations

4:27

Introduction to Systems of Linear Equations

Solving Systems of Equations - Substitution

Solving Systems of Equations - Elimination

How to Multiply Equations in Elimination Method

Classifying Systems of Linear Equations

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Solve a Linear System Graphical Approach

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Solve a Linear System Graphical Approach

Solve a Linear System û by the Substitution Method

Solve a Linear System û by the Elimination Method

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Multiple Choice

Use substitution to solve the following system of linear equations.

$4x+y=1$

$x-y=4$

Use substitution to solve the following system of linear equations.

$4x+2y=7$

$x+5y=4$

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

$2x+y=1$

$3x-y=4$

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

$10x-4y=5$

$5x-4y=1$

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

$y=5x-17$

$15x-3y=51$

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

$2x+8y=7$

$x+4y=19$

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

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Textbook Question

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

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Textbook Question

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.

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Textbook Question

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

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Textbook Question

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.

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Textbook Question

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.

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Textbook Question

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

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

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Textbook Question

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?

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Textbook Question

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

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Textbook Question

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

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

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Textbook Question

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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Textbook Question

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

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Textbook Question

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?

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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

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Textbook Question

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

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Textbook Question

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.

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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?

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Textbook Question

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

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Textbook Question

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?

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Textbook Question

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).

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Textbook Question

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).

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Textbook Question

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?

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Textbook Question

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?

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Textbook Question

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.

Solve the systems in Exercises 79–80. log x^2=y+3, log x=y−1

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Textbook Question

Solve the systems in Exercises 79–80. log_y x=3, log_y (4x)=5

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Textbook Question

Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

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Textbook Question

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

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Textbook Question

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.

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Textbook Question

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.

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. x^2+y^2+3y=22, 2x+y=−1

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. y=(x+3)^2, x+2y=−2

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. x^2+(y−2)^2=4, x^2−2y=0

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. x^3+y=0, x^2−y=0

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. x^2+4y^2=20, x+2y=6

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. 2x^2+y^2=18, xy=4

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. 3x^2+4y^2=16, 2x^2−3y^2=5

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Textbook Question

In Exercises 19–28, solve each system by the addition method. y^2−x=4, x^2+y^2=4

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Textbook Question

In Exercises 19–28, solve each system by the addition method. x^2+y^2=25, (x−8)^2+y^2=41

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Textbook Question

In Exercises 19–28, solve each system by the addition method. 3x^2+4y^2−16=0, 2x^2−3y^2−5=0

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Textbook Question

In Exercises 19–28, solve each system by the addition method. x^2−4y^2=−7, 3x^2+y^2=31

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Textbook Question

In Exercises 47–52, solve each system by the method of your choice. 3/x^2+1/y^2=7, 5/x^2−2/y^2=−3

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In Exercises 47–52, solve each system by the method of your choice. −4x+y=12, y=x^3+3x^2

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Textbook Question

In Exercises 19–28, solve each system by the addition method. x^2+y^2=13,x^2−y^2=5

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. x+y=1, (x-1)^2+(y+2)^2=10

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. x+y=1, x^2+xy-y^2=-5

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In Exercises 1–18, solve each system by the substitution method. xy=3, x^2+y^2=10

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. y^2=x^2-9, 2y=x-3

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. xy=6, 2x-y=1

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. x^2+y^2=25, x-y=1

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. y=x^2-4x-10, y=-x^2-2x+14

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x^2-4x+4

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x^2−4

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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

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Textbook Question

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 x^2 + y^2 = 25

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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 x^2 + y^2 = 10

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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)? x^2 + y = 2 (1) x - y = 0 (2)

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Textbook Question

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.

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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.

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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. 2x^2 = 3y + 23 y = 2x - 5

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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 = 3x^2 x^2 + y^2 = 10

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. x^2 - y = 0 x + y = 2

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = x^2 - 2x + 1 x - 3y = -1

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = x^2 + 6x + 9 x + 2y = -2

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = 6x + x^2 4x - y = -3

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Textbook Question

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

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. x^2 + y^2 = 10 2x^2 - y^2 = 17

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. x^2 + y^2 = 0 2x^2 - 3y^2 = 0

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. x^2 + 2y^2 = 9 x^2 + y^2 = 25

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Textbook Question

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

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. 5x^2 - 2y^2 = 25 10x^2 + y^2 = 50

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. 2xy + 1 = 0 x + 16y = 2

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. 3x^2 - y^2 = 11 xy = 12

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Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 9 to 2 and whose product is 162.

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Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 4 to 3 and are such that the sum of their squares is 100.

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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.

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Textbook Question

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

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Answer each question. A line passes through the points of intersection of the graphs of y = x^2 and x^2 + y^2 = 90. What is the equation of this line?

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Solve each problem. Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.^3 and lateral surface area 65 in.^2.

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Textbook Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q Find the equilibrium demand.

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Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q Find the equilibrium price (in dollars).

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Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q) Find the equilibrium demand.

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Textbook Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q) Find the equilibrium price (in dollars).

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Solve each problem. Find all values of b such that the straight line 3x - y = b touches the circle x^2 + y^2 = 25 at only one point.

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Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x^2 + y^2 = 20 and x^2 - y = 0.

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Textbook Question

In Exercises 47–52, solve each system by the method of your choice. 2x^2+xy=6, x^2+2xy=0

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7. Systems of Equations & Matrices

Two Variable Systems of Linear Equations

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