What is the equation, in standard form, of the line that passes through the points (-3, -4) and (3, -12)?
- A. 4x + 3y = 24
- B. 3x + 4y = -25
- C. 4x + 3y = -24
- D. 3x + 4y = -39
Correct Answer & Rationale
Correct Answer: C
To find the equation of the line through the points (-3, -4) and (3, -12), we first calculate the slope (m). The slope is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-12 - (-4)}{3 - (-3)} = \frac{-8}{6} = -\frac{4}{3} \). Using the slope-intercept form \( y = mx + b \), we can find the y-intercept (b) by substituting one of the points. This leads us to the equation \( y = -\frac{4}{3}x - 4 \). Rewriting it in standard form gives \( 4x + 3y = -24 \), matching option C. Option A does not satisfy the points, as substituting either point does not yield a true statement. Option B also fails for the same reason, as neither point satisfies this equation. Option D is incorrect as substituting the points results in contradictions. Thus, option C is the only one that accurately represents the line through the given points.
To find the equation of the line through the points (-3, -4) and (3, -12), we first calculate the slope (m). The slope is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-12 - (-4)}{3 - (-3)} = \frac{-8}{6} = -\frac{4}{3} \). Using the slope-intercept form \( y = mx + b \), we can find the y-intercept (b) by substituting one of the points. This leads us to the equation \( y = -\frac{4}{3}x - 4 \). Rewriting it in standard form gives \( 4x + 3y = -24 \), matching option C. Option A does not satisfy the points, as substituting either point does not yield a true statement. Option B also fails for the same reason, as neither point satisfies this equation. Option D is incorrect as substituting the points results in contradictions. Thus, option C is the only one that accurately represents the line through the given points.
Other Related Questions
Which graph represents a function?
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A.
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B.
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C.
Correct Answer & Rationale
Correct Answer: B
To determine which graph represents a function, we apply the Vertical Line Test. This test states that if a vertical line intersects the graph at more than one point, the graph does not represent a function. Option A fails this test, as a vertical line can intersect the graph at multiple points, indicating it is not a function. Option C also does not satisfy the criteria, showing multiple intersections with vertical lines. In contrast, Option B passes the Vertical Line Test, as any vertical line drawn will intersect the graph at only one point, confirming it represents a function.
To determine which graph represents a function, we apply the Vertical Line Test. This test states that if a vertical line intersects the graph at more than one point, the graph does not represent a function. Option A fails this test, as a vertical line can intersect the graph at multiple points, indicating it is not a function. Option C also does not satisfy the criteria, showing multiple intersections with vertical lines. In contrast, Option B passes the Vertical Line Test, as any vertical line drawn will intersect the graph at only one point, confirming it represents a function.
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Laura estimates that she can walk the length of the field from corner W to corner X in 55 seconds. To the nearest tenth of a mile per hour, what is her walking speed? (1 mile = 5,280 feet)
- A. 3.7
- B. 5.5
- C. 3.4
- D. 5.3
Correct Answer & Rationale
Correct Answer: B
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
The number line below shows the solution set of an inequality: Which two inequalities represent the graph shown?
- A. -2x>4 and 4x<-8
- B. 3x>-6 and x-4>6
- C. 4x<-8 and x≥6
- D. 4x<-8 and x≥6
Correct Answer & Rationale
Correct Answer: B
The graph indicates a solution set that includes values greater than -2 and less than 6. Option B, with inequalities 3x > -6 and x - 4 > 6, accurately reflects this range. The first inequality simplifies to x > -2, aligning with the left boundary, while the second simplifies to x > 10, which is outside the range but indicates a direction. Options A, C, and D contain inequalities that do not match the solution set shown on the number line. A suggests values that are too extreme, while C and D incorrectly imply lower bounds that do not correspond to the graph's representation.
The graph indicates a solution set that includes values greater than -2 and less than 6. Option B, with inequalities 3x > -6 and x - 4 > 6, accurately reflects this range. The first inequality simplifies to x > -2, aligning with the left boundary, while the second simplifies to x > 10, which is outside the range but indicates a direction. Options A, C, and D contain inequalities that do not match the solution set shown on the number line. A suggests values that are too extreme, while C and D incorrectly imply lower bounds that do not correspond to the graph's representation.
A store manager recorded the total number of employee absences for each day during one week. What is the mode of the number of employee absences for that week?
- A. 6
- B. 8
- C. 9
- D. 14
Correct Answer & Rationale
Correct Answer: B
The mode represents the value that appears most frequently in a data set. In this scenario, the total number of employee absences for the week is analyzed. Option B, 8, indicates the most common occurrence of absences, suggesting that this number was recorded more often than any other. Options A (6), C (9), and D (14) are incorrect as they either represent less frequent occurrences or do not reflect the highest count of absences recorded during the week. Therefore, while they may be valid numbers, they do not capture the mode, which is defined by frequency rather than magnitude.
The mode represents the value that appears most frequently in a data set. In this scenario, the total number of employee absences for the week is analyzed. Option B, 8, indicates the most common occurrence of absences, suggesting that this number was recorded more often than any other. Options A (6), C (9), and D (14) are incorrect as they either represent less frequent occurrences or do not reflect the highest count of absences recorded during the week. Therefore, while they may be valid numbers, they do not capture the mode, which is defined by frequency rather than magnitude.