Systems of Linear Equations in Two Variables

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

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. 2x + 5y = - 4 3x - y = 11

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

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 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.

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?

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. 5x + 2y = 0 x - 3y = 0

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

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

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?

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.

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

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

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

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

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

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. 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

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

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. 2x - 3y = 8 - 2x 3x + 4y = x + 3y + 14

In Exercises 19–30, solve each system by the addition method. 4x + 3y = 15 2x - 5y = 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. 3x - 2y = − 5 4x + y = 8

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

In Exercises 1–4, determine if the given ordered triple is a solution of the system. (4, 1, 2) x−2y=2, 2x+3y=11, y−4z=−7

Solve each system in Exercises 5–18. 3x+2y−3z=−2, 2x−5y+2z=−2, 4x−3y+4z=  10

Find the quadratic function y = ax^2 + bx + c whose graph passes through the points (1, 4), (3, 20), and (-2, 25).

Solve each system in Exercises 5–18. x+y=−4, y−z=1, 2x+y+3z=−21

In Exercises 19–22, find the quadratic function y = ax^2+bx+c whose graph passes through the given points. (−1,−4), (1,−2), (2, 5)

Solve by eliminating variables:

In Exercises 1–4, determine if the given ordered triple is a solution of the system. (2,−1, 3) x+  y+0z=4, x−2y−0z=1, 2x−y−2z=−1

Solve each system in Exercises 5–18. 4x−0y+2z=11, x+2y−z=−1, 2x+2y−3z=−1

Solve each system in Exercises 12–13. The is a piecewise function

Solve each system in Exercises 5–18. 2x+y=2, x+y−z=4, 3x+2y+z=0

In Exercises 19–22, find the quadratic function y = ax^2+bx+c whose graph passes through the given points. (−1, 6), (1, 4), (2, 9)

Solve each system in Exercises 25–26. (x+3)/2 − (y−1)/2 + (z+2)/4 = 3/2, (x−5)/2 + (y+1)/3 − z/4 = − 25/6, (x−3)/4 − (y+1)/2 + (z−3)/2= − 5/2

Solve each system in Exercises 5–18. x+0y+2z=11, x+0y+3z=14, x+2y−0z=5

Solve each system in Exercises 5–18. 2x−4y+3z=17, x+2y−z=0, 4x−y−z=6

Solve each system in Exercises 5–18. 3(2x+y)+5z=−1, 2(x−3y+4z)=−9, 4(1+x)=−3(z−3y)

Solve each system in Exercises 25–26. (x+2)/6 − (y+4)/3 + z/2 = 0, (x+1)/2 + (y−1)/2 − z/4 = 9/2, (x−5)/4 + (y+1)/3 + (z−2)/2 = 19/4

Exercises 57–59 will help you prepare for the material covered in the next section. Solve: A + B = 3, 2A - 2B + C = 17, 4A - 2C =14

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (3x+16)/(x + 1) (x − 2)²

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x-2)(x-3)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^2 - 3x - 4)/x(x + 2)(x - 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4x² - 7x - 3/(x^3 -x)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x²+2x+7/x(x − 1)^2

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (4x^2+3x+14)/(x^3 - 8)

Exercises 57–59 will help you prepare for the material covered in the next section. Subtract: 3/(x−4) − 2/(x+2).

Find the partial fraction decomposition of 4x²+5x-9/(x³- 6x-9)

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (6x^2-14x-27)/(x+2) (x − 3)^2

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (7x^2 -9x+3)/(x²+7)²

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (7x-4)/(x^2-x-12)

In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4x^2+13x-9/x (x − 1)(x+3)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x^2-6x+3)/(x − 2)³

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^3 + 5x^2 + 7x - 1)/(x^2 + x + 1)^2

In Exercises 9–42, write the partial fraction decomposition of each rational expression.5x^2+6x+3/(x + 1)(x² + 2x + 2)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x^3-4x^2+9x-5)/(x^2 -2x+3)^2

Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum:

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.(11x - 10)/(x − 2) (x + 1)

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x^3 + x² /(x² + 4)^2

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (3x +50)/(x -9)(x +2)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x^2 +2x -3)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (6x-11)/(x − 1)²

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (7x^2 - 7x + 23)/(x - 3)(x^2 + 4)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 5x^2 -6x+7/(x − 1) (x² + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x^3+x^2+2/(x² + 2)²

In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^4-x^2+2)/(x^3-x^2)

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 5x²-6x+7 /(x − 1) (x² + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 4/(2x^2 -5x -3)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (2x^2 -18x -12)/x³- 4x

In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - 2)(x^2 + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x^2/(x − 1)² (x + 1)

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 6x^2-x+1/(x^3 + x²+x+1)

In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^5+2)/(x^2-1)

Exercises 57–59 will help you prepare for the material covered in the next section. Add: (5x−3)/(x^2+1) + 2x/(x^2+1)^2.

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

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

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

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

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

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

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.

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

Solve the system

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

In Exercises 1–26, graph each inequality. y≤(1/3)x

In Exercises 1–26, graph each inequality. x^2+y^2≤1

In Exercises 1–26, graph each inequality. y≥x^2−9

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 2x−5y≤10, 3x−2y>6

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. −2≤x<5

In Exercises 39–45, graph each inequality. 3x - 4y > 12

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. y≥x^2−1, x−y≥−1

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.

This

is a piecewise function. Refer to the textbook.

In Exercises 1–26, graph each inequality. x−2y>10

In Exercises 1–26, graph each inequality. y>−3

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 3x+6y≤6, 2x+y≤8

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x≤2, y≥−1

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+y>4, x+y>−1

In Exercises 39–45, graph each inequality. y ≤ 2^x

In Exercises 1–26, graph each inequality. x+2y≤8

In Exercises 1–26, graph each inequality. x≤−3

In Exercises 1–26, graph each inequality. (x−2)^2+(y+1)^2<9

In Exercises 1–26, graph each inequality. y≥log_2(x+1)

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+2y≤4, y≥x−3

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+y>4, x+y<−1

In Exercises 39–45, graph each inequality. y ≤ x^2 - 1

In Exercises 1–26, graph each inequality. y>2x−1

In Exercises 1–26, graph each inequality. x^2+y^2>25

In Exercises 1–26, graph each inequality. y>2^x

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. y>2x−3, y<−x+6

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x−y≤1, x≥2

In Exercises 39–45, graph each inequality. y ≤ (-1/2)x + 2

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.

This

is a piecewise function. Refer to the textbook.

In Exercises 1–4, find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=5x+6y

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Find the value of the objective function z = 2x + 3y at each corner of the graphed region shown. What is the maximum value of the objective function? What is the minimum value of the objective function?

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

In Exercises 1–4, find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=40x+50y

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.