Two points (a,b) and (c,d) are shown on a graph. Which of the following equations correctly represents the slope of the line that passes through these points.
- A. (b-d)/(a-c)
- B. (d-b)/(c-a)
- C. (b-d)/(c-a)
- D. (d-b)/(a-c)
Correct Answer & Rationale
Correct Answer: B
To determine the slope of a line passing through two points, the formula used is \((y_2 - y_1) / (x_2 - x_1)\). In this case, for points \((a, b)\) and \((c, d)\), we can label \((x_1, y_1) = (a, b)\) and \((x_2, y_2) = (c, d)\). Option B, \((d - b) / (c - a)\), correctly applies this formula, with \(d\) as \(y_2\) and \(b\) as \(y_1\). Option A, \((b - d) / (a - c)\), incorrectly reverses the subtraction for both \(y\) and \(x\). Option C, \((b - d) / (c - a)\), misplaces the order of \(y\) values, leading to an incorrect slope sign. Option D, \((d - b) / (a - c)\), also incorrectly reverses the \(x\) values, yielding an incorrect result.
To determine the slope of a line passing through two points, the formula used is \((y_2 - y_1) / (x_2 - x_1)\). In this case, for points \((a, b)\) and \((c, d)\), we can label \((x_1, y_1) = (a, b)\) and \((x_2, y_2) = (c, d)\). Option B, \((d - b) / (c - a)\), correctly applies this formula, with \(d\) as \(y_2\) and \(b\) as \(y_1\). Option A, \((b - d) / (a - c)\), incorrectly reverses the subtraction for both \(y\) and \(x\). Option C, \((b - d) / (c - a)\), misplaces the order of \(y\) values, leading to an incorrect slope sign. Option D, \((d - b) / (a - c)\), also incorrectly reverses the \(x\) values, yielding an incorrect result.
Other Related Questions
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.
Tina Is designing a cabin. One of her plans for the cabin is a rectangle twice as long as it is wide, with 10 feet (ft) of the length reserved for the Kitchen and the bathroom. The diagram shows this basic plan. Tina wants the area of the main room to be 300 square feet. Which equation can be used to find x, the width, in feet, of the main room?
- A. 2x^2 + 10x - 300 = 0
- B. 2x^2 - 10x - 300 = 0
- C. 2x^2 - 20x - 300 = 0
- D. 2x^2 + 20x - 300 = 0
Correct Answer & Rationale
Correct Answer: B
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
A landscape worker is building a rock wall around a triangular flower garden. He has completed the rock wall on two sides of the garden.
The perimeter of the garden is 239 feet. What is the length, in feet, of the rock wall that the worker still needs to complete?
- A. 101
- B. 185
- C. 54
- D. 138
Correct Answer & Rationale
Correct Answer: D
To determine the length of the rock wall still needed, first, the total perimeter of the triangular garden is 239 feet. The worker has already completed two sides, leaving one side to be built. To find the length of the remaining side, we subtract the lengths of the two completed sides from the total perimeter. The answer of 138 feet indicates that the lengths of the two sides combined equal 101 feet (239 - 138 = 101). Option A (101) represents the combined length of the two completed sides, not the remaining side. Option B (185) exceeds the total perimeter, which is impossible. Option C (54) does not fit the calculations based on the perimeter. Thus, only option D accurately reflects the length of the remaining side to complete the wall.
To determine the length of the rock wall still needed, first, the total perimeter of the triangular garden is 239 feet. The worker has already completed two sides, leaving one side to be built. To find the length of the remaining side, we subtract the lengths of the two completed sides from the total perimeter. The answer of 138 feet indicates that the lengths of the two sides combined equal 101 feet (239 - 138 = 101). Option A (101) represents the combined length of the two completed sides, not the remaining side. Option B (185) exceeds the total perimeter, which is impossible. Option C (54) does not fit the calculations based on the perimeter. Thus, only option D accurately reflects the length of the remaining side to complete the wall.
A cyclist can travel 17.6 feet per second. The cyclist would have a better understanding of her speed if it were measured in miles per hour. Which of these completes the expression used to convert the speed of the cyclist to miles per hour?
- A. 1 hour/60 seconds = 1 mile/5,280 feet
- B. 60 minutes/1 hour = 1 mile/5280 feet
- C. 60 minutes/1 hour = 5280 feet/1 mile
- D. 12 inches/1 foot = 60 minutes/1 hour
Correct Answer & Rationale
Correct Answer: C
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.