Systems of Linear Inequalities in Two Variables Practice Questions — SAT — Page 3

Question 41Question →

A nutritionist is designing a meal plan containing $x$ grams of protein and $y$ grams of carbohydrates. The meal plan must satisfy the following system of inequalities:

\begin{aligned} y &\ge 1.5x + 15 \\ 3x + 2y &\le 120 \end{aligned}

What is the maximum possible number of grams of protein, $x$ , that can be included in the meal plan?

Show answer & explanation

Answer: 15

Answer

The maximum possible number of grams of protein that can be included is 15.

To find the maximum possible value of

x

, we determine the region defined by the system of inequalities. The system restricts the values to the region above the line

y = 1.5x + 15

and below the line

3x + 2y = 120

. Since the first inequality limits

y

from below and the second limits

y

from above, the feasible region narrows as

x

increases, terminating at the intersection of the two boundary lines. Substituting

y = 1.5x + 15

into

3x + 2y = 120

gives

3x + 2(1.5x + 15) = 120

. Simplifying this yields

3x + 3x + 30 = 120

, which simplifies further to

6x = 90

, giving

x = 15

. Thus, the maximum value of

x

is 15.

Step-by-Step Solution

1

Identify the boundary lines of the system of inequalities.

The boundary lines are

y = 1.5x + 15

and

3x + 2y = 120

.

The maximum value of

x

under these linear constraints occurs at the intersection of the boundary lines of the feasible region.

2

Substitute the expression for

y

from the first boundary equation into the second equation.

3x + 2(1.5x + 15) = 120

This allows us to solve for

x

by eliminating

y

.

3

Simplify the equation and solve for

x

.

3x + 3x + 30 = 120 \implies 6x + 30 = 120 \implies 6x = 90 \implies x = 15

.

Solving the linear equation gives the

x

-coordinate of the intersection point.

4

Verify that this point lies in the feasible region and represents the maximum possible value of

x

.

At

x=15

,

y=37.5

. Since

y \ge 1.5x + 15

restricts the region above the line and

3x + 2y \le 120

restricts it below the line, the region lies to the left of the intersection point

(15, 37.5)

. Thus, the maximum value of

x

is 15.

Confirming the geometry of the feasible region ensures the intersection point is indeed the maximum value.

Key Concept

Solving systems of linear inequalities to find the boundaries and extreme values of a feasible region.

Question 42Question →

Consider the system of inequalities below:

\begin{aligned} 3x + y &> 5 \\ x - 2y &> 4 \end{aligned}

Which of the following coordinate pairs $(x, y)$ is a solution to the system?

(3, 1)

(2, -1)

(4, -2)

(3, 2)

Show answer & explanation

Answer: $(4, -2)$

Answer

(4, -2)

The coordinate pair

(4, -2)

is the correct answer because substituting these values into both inequalities of the system produces true statements:

3(4) + (-2) = 10 > 5

and

4 - 2(-2) = 8 > 4

.

Step-by-Step Solution

1

Substitute the coordinates of the candidate point into the first inequality,

3x + y > 5

.

For

(4, -2)

, we get

3(4) + (-2) = 12 - 2 = 10

. Since

10 > 5

, the first inequality is satisfied.

A coordinate pair must satisfy both inequalities in the system to be a solution.

2

Substitute the coordinates of the candidate point into the second inequality,

x - 2y > 4

.

For

(4, -2)

, we get

4 - 2(-2) = 4 + 4 = 8

. Since

8 > 4

, the second inequality is also satisfied.

Since both inequalities are true for

(4, -2)

, it is a valid solution to the system.

Key Concept

A coordinate pair

(x, y)

is a solution to a system of linear inequalities if and only if it makes all inequalities in the system true when substituted.

Estimated Time:1m 30s

Question 43Question →

A software programmer is writing test cases for a new application. The programmer must write at least 15 test cases in total, consisting of $x$ unit tests and $y$ integration tests. Each unit test takes 10 minutes to write, and each integration test takes 30 minutes to write. If the programmer has at most 300 minutes to write all the test cases, what is the maximum number of integration tests the programmer can write?

Show answer & explanation

Answer: 7

Answer

7

Step-by-Step Solution

1

Set up the system of inequalities representing the given constraints.

x + y \ge 15

and

10x + 30y \le 300

The total number of tests must be at least 15, and the total time taken by writing

x

unit tests (10 minutes each) and

y

integration tests (30 minutes each) cannot exceed 300 minutes.

2

Simplify the time inequality and combine it with the total test count constraint to isolate

y

.

x + 3y \le 30

. Substituting

x \ge 15 - y

into this inequality yields

(15 - y) + 3y \le 30 \implies 15 + 2y \le 30

.

Simplification and substitution help find the upper bound for the number of integration tests.

3

Solve for

y

and determine the maximum integer value.

2y \le 15 \implies y \le 7.5

. The largest integer satisfying this inequality is 7.

The number of integration tests must be a whole number, so we round down to the nearest integer.

4

Verify that a valid integer number of unit tests (

x

) exists when

y = 7

.

When

y = 7

, we get

x \ge 15 - 7 \implies x \ge 8

and

x + 3(7) \le 30 \implies x \le 9

. The integers

x = 8

and

x = 9

both satisfy the conditions.

We must confirm that the maximum value of

y

is achievable with an integer number of unit tests.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 44Question →

In the $xy$ -plane, a system of inequalities consists of the following:

y > -\frac{1}{2}x + 2

y \leq 3x - 1

Which of the following coordinate pairs $(x, y)$ is a solution to this system?

(1, 1)

(2, 2)

(1, 4)

(2, 0)

Show answer & explanation

Answer: $(2, 2)$

Answer

The coordinate pair

(2, 2)

The coordinate pair

(2, 2)

is the correct answer because substituting

x = 2

and

y = 2

into both inequalities yields true statements. For the first inequality,

2 > -\frac{1}{2}(2) + 2

simplifies to

2 > 1

, which is true. For the second inequality,

2 \leq 3(2) - 1

simplifies to

2 \leq 5

, which is also true. Since the point satisfies both inequalities, it lies in the solution region.

Step-by-Step Solution

1

Understand the definition of a solution to a system of inequalities.

A coordinate pair

(x, y)

is a solution to a system of inequalities if and only if it satisfies both inequalities simultaneously when substituted.

This establishes the verification method for checking the options.

2

Substitute the coordinate pair

(2, 2)

into the first inequality:

y > -\frac{1}{2}x + 2

.

2 > -\frac{1}{2}(2) + 2 \implies 2 > -1 + 2 \implies 2 > 1

.

This determines if the coordinate pair satisfies the first boundary condition.

3

Substitute the coordinate pair

(2, 2)

into the second inequality:

y \leq 3x - 1

.

2 \leq 3(2) - 1 \implies 2 \leq 6 - 1 \implies 2 \leq 5

.

This determines if the coordinate pair satisfies the second boundary condition.

4

Conclude whether both statements are true.

Since

2 > 1

is true and

2 \leq 5

is true, the coordinate pair

(2, 2)

is a solution to the system.

Both conditions must be met for the coordinate pair to belong to the solution set.

Key Concept

Verifying coordinate solutions for systems of linear inequalities

Estimated Time:1m 30s

Question 45Question →

A point $(x, y)$ in the coordinate plane satisfies the system of inequalities below.

y \geq 2x - 4

y \leq -x + 5

x \geq 0

y \geq 0

What is the maximum possible value of $x$ for this point?

Show answer & explanation

Answer: 3

Answer

The maximum possible value of

x

that satisfies the system of inequalities is 3.

To find the maximum possible value of

x

for a point satisfying the system of inequalities, we identify the vertices of the region. The upper-right boundary of the region is formed by the intersection of the lines

y = 2x - 4

and

y = -x + 5

. Solving

2x - 4 = -x + 5

gives

3x = 9 \implies x = 3

. At this point,

y = -3 + 5 = 2

, which satisfies the constraints

x \geq 0

and

y \geq 0

. The other boundary vertices of the region are

(0, 0)

,

(2, 0)

, and

(0, 5)

. Comparing the x-coordinates of these vertices (

0

,

2

, and

3

), we see that the maximum possible value of

x

is

3

.

Step-by-Step Solution

1

Find the intersection point of the boundary lines

y = 2x - 4

and

y = -x + 5

.

x = 3

Setting the two boundary line equations equal to each other (

2x - 4 = -x + 5

) allows us to find the x-coordinate where the boundaries cross.

2

Substitute

x = 3

back into one of the boundary equations to find the y-coordinate.

y = 2

This yields the intersection point

(3, 2)

which lies on both boundary lines.

3

Verify that the intersection point

(3, 2)

satisfies the other constraints:

x \geq 0

and

y \geq 0

.

3 \geq 0

and

2 \geq 0

(both true)

The point must lie within the first quadrant to be a valid solution.

4

Determine the remaining boundary vertices of the solution region in the first quadrant.

Vertices are

(0, 0)

,

(0, 5)

,

(2, 0)

, and

(3, 2)

.

Comparing all vertices will confirm if

(3, 2)

indeed provides the maximum value of

x

.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 46Question →

A logistics company uses a delivery drone to transport two types of packages. Let $x$ represent the number of Type A packages and $y$ represent the number of Type B packages in a single flight. The drone can carry at most 12 packages in total. Additionally, to balance the drone, the total weight of the cargo must be at least 15 pounds. Each Type A package weighs 2 pounds, and each Type B package weighs 1 pound. Which of the following combinations of Type A and Type B packages is a viable shipment for the drone?

3 Type A packages and 8 Type B packages

4 Type A packages and 5 Type B packages

5 Type A packages and 6 Type B packages

6 Type A packages and 7 Type B packages

Show answer & explanation

Answer: 5 Type A packages and 6 Type B packages

Answer

5 Type A packages and 6 Type B packages

The correct combination of 5 Type A packages and 6 Type B packages satisfies all system requirements. The total package count of 11 is less than or equal to the drone's limit of 12 packages (

5 + 6 \leq 12

). Furthermore, the total weight of 16 pounds meets the minimum balance requirement of 15 pounds (

2(5) + 6 \geq 15

).

Step-by-Step Solution

1

Write the system of linear inequalities that represents the given constraints.

The capacity constraint is

x + y \leq 12

, and the weight constraint is

2x + y \geq 15

, where

x

and

y

are non-negative integers.

To define the mathematical boundaries for a valid shipment.

2

Substitute the package counts from the correct combination into both inequalities to verify they are satisfied.

For 5 Type A packages (

x = 5

) and 6 Type B packages (

y = 6

):
- Package count:

5 + 6 = 11 \leq 12

(True)
- Cargo weight:

2(5) + 6 = 16 \geq 15

(True)

To confirm that the chosen combination satisfies all system requirements.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 30s

Question 47Question →

In the $xy$ -plane, a point with coordinates $(x, y)$ lies in the region defined by the system of inequalities below.

y + 2x \leq 12

x - 2y \leq 6

x \geq 2

What is the maximum possible value of $y$ ?

Show answer & explanation

Answer: 8

Answer

8

The system of inequalities bounds the solution set. The upper boundary is given by

y \leq -2x + 12

. Because the slope is negative, the maximum value of

y

on this boundary occurs at the smallest possible value of

x

. The constraint

x \geq 2

dictates that the minimum value of

x

is 2. Substituting

x = 2

into the boundary equation yields

y = -2(2) + 12 = 8

. Checking this coordinate against the third inequality,

2 - 2(8) = -14 \leq 6

, verifies that

(2, 8)

is a valid solution.

Step-by-Step Solution

1

Express the first two inequalities in terms of y.

y \leq -2x + 12

and

y \geq \frac{1}{2}x - 3

This helps identify the upper and lower boundaries of the solution region.

2

Determine the boundary line that limits the maximum values of y.

The upper boundary is the line

y = -2x + 12

.

Since the inequality is

y \leq -2x + 12

, any solution must lie on or below this line.

3

Find the maximum value of y on this boundary line given the constraint

x \geq 2

.

y \leq -2(2) + 12 = 8

Since the slope of the boundary line is negative, y is maximized when x is at its minimum value, which is 2.

4

Verify that the point

(2, 8)

satisfies the inequality

x - 2y \leq 6

.

2 - 2(8) = -14 \leq 6

, which is true.

This confirms that the point

(2, 8)

is indeed in the solution set of the system.

Key Concept

Maximizing a coordinate value subject to a system of linear inequalities in two variables

Alternative Method

Instead of graphing or checking boundaries, we can algebraically solve for the boundary. Since

x \geq 2

, multiplying by

-2

and reversing the inequality gives

-2x \leq -4

. Adding 12 to both sides gives

-2x + 12 \leq 8

. Since

y \leq -2x + 12

, we get

y \leq 8

. Checking if

y = 8

and

x = 2

satisfies the second inequality

x - 2y \leq 6

confirms that

2 - 16 = -14 \leq 6

is true, meaning

y = 8

is indeed a valid solution and thus the maximum.

Estimated Time:1m 30s

Question 48Question →

An artist creates $x$ small sculptures and $y$ large sculptures. Each small sculpture requires 3 hours of crafting and 2 hours of painting. Each large sculpture requires 7 hours of crafting and 3 hours of painting. The artist can spend at most 120 hours on crafting and at most 50 hours on painting. Which of the following pairs of small and large sculptures can the artist create under these constraints?

5 small sculptures and 15 large sculptures

12 small sculptures and 9 large sculptures

10 small sculptures and 8 large sculptures

15 small sculptures and 12 large sculptures

Show answer & explanation

Answer: 10 small sculptures and 8 large sculptures

Answer

10 small sculptures and 8 large sculptures

The option featuring 10 small sculptures and 8 large sculptures is correct because substituting

x = 10

and

y = 8

satisfies both linear inequalities representing the constraints. Specifically, the crafting time of 86 hours is less than or equal to the maximum allowed 120 hours (

3(10) + 7(8) = 86 \leq 120

), and the painting time of 44 hours is less than or equal to the maximum allowed 50 hours (

2(10) + 3(8) = 44 \leq 50

).

Step-by-Step Solution

1

Set up the system of inequalities representing the constraints.

The crafting constraint is

3x + 7y \leq 120

. The painting constraint is

2x + 3y \leq 50

. The variables must be non-negative:

x \geq 0

and

y \geq 0

.

To model the limits on crafting hours and painting hours mathematically.

2

Substitute the values of each option into the system to verify which one satisfies both inequalities.

For the option with 10 small sculptures and 8 large sculptures (

x = 10, y = 8

):

3(10) + 7(8) = 30 + 56 = 86 \leq 120

(True)

2(10) + 3(8) = 20 + 24 = 44 \leq 50

(True)

Only a coordinate pair that makes both inequality statements true is a valid solution.

Key Concept

Systems of Linear Inequalities in Two Variables

Systems of Linear Inequalities in Two Variables

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Step-by-Step Solution

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Common Mistakes

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Step-by-Step Solution

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Common Mistakes

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Step-by-Step Solution

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Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes