Section 3.4 – Equivalent Linear Relations

1) Determine the x-intercept of 2x - 4y = 6

-3

-3/2

3/2

3

The x-intercept is the point where the line crosses the x-axis. To find the x-intercept, we set y = 0 and solve for x. Plugging in y = 0 into the equation 2x - 4y = 6, we get 2x - 4(0) = 6. Simplifying this equation, we get 2x = 6. Dividing both sides by 2, we find that x = 3. Therefore, the x-intercept of the equation 2x - 4y = 6 is 3.

Explanation

The x-intercept is the point where the line crosses the x-axis. To find the x-intercept, we set y = 0 and solve for x. Plugging in y = 0 into the equation 2x - 4y = 6, we get 2x - 4(0) = 6. Simplifying this equation, we get 2x = 6. Dividing both sides by 2, we find that x = 3. Therefore, the x-intercept of the equation 2x - 4y = 6 is 3.

2) Locate three points on the line 8x - 2y = 16

(0, 2), (-8, 0) and (-12, -1)

(0, -8), (2, 0) and (-1, -12)

(0, 8), (-2, 0) and (-1, -12)

(0, -2), (8, 0) and (-12, -1)

The given equation is in the form of a linear equation, where x and y are variables. To locate points on the line, we can substitute different values of x and solve for y. By substituting x = 0 into the equation, we get -2y = 16, which gives y = -8. Therefore, the point (0, -8) lies on the line. Similarly, substituting x = 2 and x = -1 into the equation gives y = 0 and y = -12 respectively, which means that (2, 0) and (-1, -12) are also points on the line. Hence, the answer is (0, -8), (2, 0) and (-1, -12).

Explanation

The given equation is in the form of a linear equation, where x and y are variables. To locate points on the line, we can substitute different values of x and solve for y. By substituting x = 0 into the equation, we get -2y = 16, which gives y = -8. Therefore, the point (0, -8) lies on the line. Similarly, substituting x = 2 and x = -1 into the equation gives y = 0 and y = -12 respectively, which means that (2, 0) and (-1, -12) are also points on the line. Hence, the answer is (0, -8), (2, 0) and (-1, -12).

3) Mary works part time at a clothing store and makes $240 in a week. She earns 4% commision on her shirt sales and 6% commission on her jeans sales. What would be the equation that would represent this situation? (Let "s" represents the number of shirts she sells and "j" represents the number of jeans she sells)

4s + 6j = 240

0.4s + 0.6j = 240

0.04s + 0.06j = 240

None of the above

The given equation, 0.04s + 0.06j = 240, represents the situation accurately. The equation shows that Mary earns a 4% commission on the number of shirts she sells (0.04s) and a 6% commission on the number of jeans she sells (0.06j). The sum of these commissions should equal her total earnings, which is $240. Therefore, this equation correctly models the situation described in the question.

Explanation

The given equation, 0.04s + 0.06j = 240, represents the situation accurately. The equation shows that Mary earns a 4% commission on the number of shirts she sells (0.04s) and a 6% commission on the number of jeans she sells (0.06j). The sum of these commissions should equal her total earnings, which is $240. Therefore, this equation correctly models the situation described in the question.

4) Determine the y-intercept of -5x + 7y = 8

8/5

8/7

-8/7

-8/5

The y-intercept of a linear equation is the point where the line intersects the y-axis. To determine the y-intercept of the equation -5x + 7y = 8, we need to set x = 0 and solve for y. When x = 0, the equation becomes 7y = 8. Dividing both sides by 7 gives us y = 8/7. Therefore, the y-intercept of the equation is 8/7.

Explanation

The y-intercept of a linear equation is the point where the line intersects the y-axis. To determine the y-intercept of the equation -5x + 7y = 8, we need to set x = 0 and solve for y. When x = 0, the equation becomes 7y = 8. Dividing both sides by 7 gives us y = 8/7. Therefore, the y-intercept of the equation is 8/7.

5) Petra has $7.75 in quarters and dimes. Which relation models this example?

15q + 25d = 7.75

28q + 8d = 7.75

10q + 35d = 7.75

.25q + .10d = 7.75

The correct answer is .25q + .10d = 7.75. This equation represents the relationship between the number of quarters (q) and dimes (d) that Petra has, and the total amount of money she has, which is $7.75. The coefficients of .25 and .10 represent the values of a quarter and a dime respectively, and when multiplied by the number of quarters and dimes, they give the total value of the coins. The equation accurately represents the given information and is therefore the correct relation for this example.

Explanation

The correct answer is .25q + .10d = 7.75. This equation represents the relationship between the number of quarters (q) and dimes (d) that Petra has, and the total amount of money she has, which is $7.75. The coefficients of .25 and .10 represent the values of a quarter and a dime respectively, and when multiplied by the number of quarters and dimes, they give the total value of the coins. The equation accurately represents the given information and is therefore the correct relation for this example.

6) Determine the x/y intercepts of the relation 6.25x + 3.75y = 18.75

X-intercept = 3, y-intercept = 5

X-intercept = 5, y-intercept = 3

X-intercept = -3, y-intercept = 5

X-intercept = -3, y-intercept = -5

To find the x-intercept, we set y = 0 and solve for x. Plugging in y = 0 into the equation 6.25x + 3.75y = 18.75, we get 6.25x + 0 = 18.75. Simplifying this equation gives us 6.25x = 18.75, and dividing both sides by 6.25 gives x = 3. Therefore, the x-intercept is 3.

To find the y-intercept, we set x = 0 and solve for y. Plugging in x = 0 into the equation 6.25x + 3.75y = 18.75, we get 0 + 3.75y = 18.75. Simplifying this equation gives us 3.75y = 18.75, and dividing both sides by 3.75 gives y = 5. Therefore, the y-intercept is 5.

Explanation

To find the x-intercept, we set y = 0 and solve for x. Plugging in y = 0 into the equation 6.25x + 3.75y = 18.75, we get 6.25x + 0 = 18.75. Simplifying this equation gives us 6.25x = 18.75, and dividing both sides by 6.25 gives x = 3. Therefore, the x-intercept is 3.

To find the y-intercept, we set x = 0 and solve for y. Plugging in x = 0 into the equation 6.25x + 3.75y = 18.75, we get 0 + 3.75y = 18.75. Simplifying this equation gives us 3.75y = 18.75, and dividing both sides by 3.75 gives y = 5. Therefore, the y-intercept is 5.

7) Locate three points on the line -6x + 4y - 12 = 0

(0, -2), (3, 0) and (6, 12)

(0, 2), (-3, 0), and (-12, 6)

(0, 3), (-2, 0), and (6,12)

(0, -3), (2, 0), and (12, 6)

The equation -6x + 4y - 12 = 0 represents a line in the coordinate plane. To locate points on this line, we can substitute the x and y values from each given option into the equation and check if the equation holds true. For the option (0, 3), (-2, 0), and (6,12), we can substitute the x and y values into the equation: -6(0) + 4(3) - 12 = 0, -6(-2) + 4(0) - 12 = 0, and -6(6) + 4(12) - 12 = 0. In all three cases, the equation holds true, confirming that these points lie on the line.

Explanation

The equation -6x + 4y - 12 = 0 represents a line in the coordinate plane. To locate points on this line, we can substitute the x and y values from each given option into the equation and check if the equation holds true. For the option (0, 3), (-2, 0), and (6,12), we can substitute the x and y values into the equation: -6(0) + 4(3) - 12 = 0, -6(-2) + 4(0) - 12 = 0, and -6(6) + 4(12) - 12 = 0. In all three cases, the equation holds true, confirming that these points lie on the line.

8) Mary works part time at a clothing store and makes $240 in a week. She earns 4% commision on her shirt sales and 6% commission on her jeans sales. What is the most shirt sales she could complete in one week? (Let "s" represents the number of shirts she sells and "j" represents the number of jeans she sells)

$240

$6000

$4000

None of the above

not-available-via-ai

Explanation

not-available-via-ai

9)

Which graph represents the graph of the relation 3x - 4y = 12 using the x and y interecepts?

Top-Left

Top-Right

Bottom-Left

Bottom-Right

The top-left graph represents the graph of the relation 3x - 4y = 12 using the x and y intercepts. This can be determined by analyzing the points where the graph intersects the x and y axes. The top-left graph intersects the x-axis at x = 4 and the y-axis at y = -3, which satisfies the equation 3x - 4y = 12. Therefore, the top-left graph is the correct representation of the given relation.

Explanation

The top-left graph represents the graph of the relation 3x - 4y = 12 using the x and y intercepts. This can be determined by analyzing the points where the graph intersects the x and y axes. The top-left graph intersects the x-axis at x = 4 and the y-axis at y = -3, which satisfies the equation 3x - 4y = 12. Therefore, the top-left graph is the correct representation of the given relation.

10) Write the equation of the line whose x-int= -3 and y-int = 5 in the specific form of Ax + By = C

Y = 5/3x + 5

3x - 5y = -25

3x - 5y = -15

5x - 3y = -15

The equation of a line in the specific form of Ax + By = C represents a line with coefficients A, B, and C. In the given answer, the equation 5x - 3y = -15 satisfies this form. The coefficients are A=5, B=-3, and C=-15. Therefore, the equation 5x - 3y = -15 is the correct answer.

Explanation

The equation of a line in the specific form of Ax + By = C represents a line with coefficients A, B, and C. In the given answer, the equation 5x - 3y = -15 satisfies this form. The coefficients are A=5, B=-3, and C=-15. Therefore, the equation 5x - 3y = -15 is the correct answer.

11) Joe is buying cheese and crackers for the party. If a block of cheese costs $4 and a package of crackers costs $2.50, how many of each did he buy if he spent $47 on 14 items.

8 crackers and 6 blocks of cheese

6 crackers and 8 blocks of cheese

7 crackers and 7 blocks of cheese

9 crackers and 5 blocks of cheese

Joe spent a total of $47 on 14 items. Let's assume he bought x blocks of cheese and y packages of crackers. The cost of x blocks of cheese would be 4x dollars and the cost of y packages of crackers would be 2.50y dollars. Since Joe bought a total of 14 items, we can write the equation x + y = 14. Additionally, the total cost of cheese and crackers is $47, so we can write the equation 4x + 2.50y = 47. By solving these two equations simultaneously, we find that x = 8 and y = 6, which means Joe bought 6 crackers and 8 blocks of cheese.

Explanation

Joe spent a total of $47 on 14 items. Let's assume he bought x blocks of cheese and y packages of crackers. The cost of x blocks of cheese would be 4x dollars and the cost of y packages of crackers would be 2.50y dollars. Since Joe bought a total of 14 items, we can write the equation x + y = 14. Additionally, the total cost of cheese and crackers is $47, so we can write the equation 4x + 2.50y = 47. By solving these two equations simultaneously, we find that x = 8 and y = 6, which means Joe bought 6 crackers and 8 blocks of cheese.