5. Systems of Equations and Inequalities

Question 1

A solution to a system of linear equations in two variables is an ordered pair that _____.

Answer 1

satisfies both equations in the system

Question 2

When solving a system of linear equations by graphing, the system’s solution is determined by locating _____.

Answer 2

the intersection point

Question 3

When solving

{ 3x – 2y = 5
{ y = 3x – 3

by the substitution method, we obtain x = 1/3, so the solution set is _____.

Answer 3

{(1/3, –2)}

Question 4

When solving

{ 2x + 10y = 9
{ 8x + 5y = 7

by the addition method, we can eliminate y by multiplying the second equation by _____ and then adding the equations.

Answer 4

–2

Question 5

When solving

{ 4x – 3y = 15
{ 3x – 2y = 10

by the addition method, we can eliminate y by multiplying the first equation by 2 and the second equation by _____ and then adding the equations.

Answer 5

–3

Question 6

When solving

{ 12x – 21y = 24
{ 4x – 7y = 7

by the addition method, we obtain 0 = 3, so the solution set is _____.

The linear system is a/an _____.

If you attempt to solve such a system by graphing, you will obtain two lines that are _____.

Answer 6

∅

inconsistent

parallel

Question 7

When solving

{ x = 3y + 2
{ 5x – 15y = 10

by the substitution method, we obtain 10 = 10, so the solution set is _____.

The equations in this system are called ______.

If you attempt to solve such a system by graphing, you will obtain two lines that ______.

Answer 7

{(x, y) | x = 3y + 2}
or
{(x, y) | 5x – 15 = 10}

dependent

are identical or coincide

Question 8

A company’s _____ function is the money generated by selling x units of its product.

The difference between this function and the company’s cost function is called its _____ function.

Answer 8

revenue

profit

Question 9

A company has a graph that shows the money it generates by selling x units of its product.

It also has a graph that shows its cost of producing x units of its product.

The point of intersection of these graphs is called the company’s _____.

Answer 9

break-even point

Question 10

A solution of a system of linear equations in three variables is an ordered _____ of real numbers that satisfies all?/some? (choose the correct choice) of the equations in the system.

Answer 10

triple

all

Question 11

Consider the following system:

{ x + y – z = –1
{ 2x – 2y – 5z = 7
{ 4x + y – 2z = 7

We can eliminate x from Equations 1 and 2 by multiplying Equation 1 by _____ and adding equations.

We can eliminate x from Equations 1 and 3 by multiplying Equation 1 by _____ and adding equations.

Answer 11

–2

–4

Question 12

Consider the following system:

{ x + y + z = 2
{ 2x – 3y = 3
{ 10y – z = 12

Equation 2 does not contain the variable _____.

To obtain a second equation that does not contain this variable, we can ______.

Answer 12

z

add Equations 1 and 3

Question 13

Correct or incorrect:

[7x / (x + 2)(x – 3)] = [A / (x + 2)] + [B / (x – 3)]

Answer 13

Correct

Question 14

Correct or incorrect

[3x / (x+5)(x – 4)²] = [A / (x + 5)] + [B / (x – 3)

Answer 14

Incorrect

Question 15

Correct or incorrect

[1 / (x + 1)(x² + 4)] = [A / (x + 1)] + [B / (x² + 4)]

Answer 15

Incorrect

Question 16

Correct or incorrect

[(7x – 5) / (x² + x + 1)²] = [(Ax + B) / (x² + x + 1)] + [(Cx + D) / (x² + x + 1)²]

Answer 16

Correct

Question 17

A system of two equations in two variables that contains at least one equation that cannot be expressed in the form Ax + By = C is called a system of _____ equations.

Answer 17

nonlinear

Question 18

When solving

{ x² – 4y = 4
{ x + y = –1

by the substitution method, we obtain x = –4 or x = 0, so the solution set is _____.

Answer 18

{(–4, 3), (0, 1)}

Question 19

When solving

{ 3x² + 2y² = 35
{ 4x² + 3y² = 48

by the addition method, we can eliminate x² by multiplying the first equation by –4 and the second equation by _____ and then adding the equations.

Answer 19

3

Question 20

When solving

{ x² + 4y² = 16
{ x² – y² = 1

by the addition method, we obtain y² = 3, so the solution set is _____.

Answer 20

{(2, √3), (2, –√3), (–2, √3), (–2, –√3)},

Question 21

When solving

{ x² + y² = 13
{ x² – y = 7

by the addition method, we can eliminate x² by multiplying the second equation by _____ and then adding the equations.

We obtain ______, a quadratic equation in y.

Answer 21

–1

y² + y = 6

Question 22

When solving

{ x² + 4y² = 20
{ xy = 4

by the substitution method, we can eliminate y by solving the second equation for y. We obtain y = _____.

Then we substitute _____ for _____ in the first equation.

Answer 22

4 / x

4 / x

y

Question 23

The ordered pair (3, 2) is a/an _____ of the inequality x + y > 1 because when 3 is substituted for _____ and 2 is substituted for _____, the true statement ______ is obtained.

Answer 23

solution

x

y

5 > 1

Question 24

The set of all points that satisfy a linear inequality in two variables is called the _____ of the inequality.

Answer 24

graph

Question 25

The set of all points on one side of a line is called an/an ______.

Answer 25

half-plane

Question 26

True or false

The graph of 2x – 3y > 6 includes the line 2x – 3y = 6.

Answer 26

false

Question 27

True or false

The graph of the linear equation 2x – 3y = 6 is used to graph the linear inequality 2x – 3y > 6.

Answer 27

true

Question 28

True or false

When graphing 4x – 2y ≥ 8, to determine which side of the line to shade, choose a test point on 4x – 2y = 8

Answer 28

false

Question 29

When graphing x² + y² > 25, to determine whether to shade the region inside the circle or the region outside the circle, we can use _____ as a test point.

Answer 29

(0, 0), although answers will vary

Question 30

The solution set of the system

{ x – y < 1
{ 2x + 3y ≥ 12

is the set of ordered pairs that satisfies _____ and ______.

Answer 30

x – y < 1

2x + 3y ≥ 12

Question 31

True or false

The graph of the solution set of the system

{ x – 3y < 6
{ 2x + 3y ≥ –6

includes the intersection point of x – 3y = 6 and 2x + 3y = –6.

Answer 31

false

Question 32

A method for finding the maximum or minimum value of a quantity that is subject to various limitations is called ______.

Answer 32

linear programming

Question 33

An algebraic expression in two or more variables describing a quantity that must be maximized or minimized is called a/an ______ function.

Answer 33

objective

Question 34

A system of linear inequalities is used to represent restrictions, or _____, on a function that must be maximized or minimized.

Using the graph of such a system of inequalities, the maximum and minimum values of the function occur at one or more of the _____ points.

Answer 34

constraints

corner