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
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.
Solve the equation for x: (2x-3)/5 = x/10
- A. 2
- B. 3
- C. 1\5
- D. 10
Correct Answer & Rationale
Correct Answer: A
To solve the equation \((2x-3)/5 = x/10\), first eliminate the fractions by multiplying both sides by 10, resulting in \(2(2x - 3) = x\). Simplifying gives \(4x - 6 = x\). Rearranging leads to \(4x - x = 6\), or \(3x = 6\), giving \(x = 2\). Option B (3) does not satisfy the equation when substituted back. Option C (1/5) results in a negative left side, while Option D (10) leads to an incorrect balance in the original equation. Thus, the only solution that holds true is \(x = 2\).
To solve the equation \((2x-3)/5 = x/10\), first eliminate the fractions by multiplying both sides by 10, resulting in \(2(2x - 3) = x\). Simplifying gives \(4x - 6 = x\). Rearranging leads to \(4x - x = 6\), or \(3x = 6\), giving \(x = 2\). Option B (3) does not satisfy the equation when substituted back. Option C (1/5) results in a negative left side, while Option D (10) leads to an incorrect balance in the original equation. Thus, the only solution that holds true is \(x = 2\).
A bag of dog food weighs 40 pounds. The amount of food in the bag is more than 3 times the amount needed to feed a dog for one week. Which inequality can be used to determine the possible values for p, the pounds of food needed to feed the dog for one week?
- A. p < 3(40)
- B. 3p < 40
- C. p > 3(40)
- D. 3p > 40
Correct Answer & Rationale
Correct Answer: D
To find the amount of food needed for one week, we know that the total weight of the dog food (40 pounds) is more than three times the weekly requirement (3p). Therefore, the relationship can be expressed as 3p < 40, indicating that the total food exceeds three times the weekly amount. Option A (p < 3(40)) incorrectly suggests that the weekly requirement is less than three times the total weight, which is not supported by the problem statement. Option B (3p < 40) misrepresents the relationship, as it implies the total food is less than three times the weekly need, contradicting the given information. Option C (p > 3(40)) inaccurately states that the weekly requirement exceeds three times the total weight, which is impossible given the context. Thus, the correct inequality is 3p > 40, indicating the total food is indeed more than three times the weekly requirement.
To find the amount of food needed for one week, we know that the total weight of the dog food (40 pounds) is more than three times the weekly requirement (3p). Therefore, the relationship can be expressed as 3p < 40, indicating that the total food exceeds three times the weekly amount. Option A (p < 3(40)) incorrectly suggests that the weekly requirement is less than three times the total weight, which is not supported by the problem statement. Option B (3p < 40) misrepresents the relationship, as it implies the total food is less than three times the weekly need, contradicting the given information. Option C (p > 3(40)) inaccurately states that the weekly requirement exceeds three times the total weight, which is impossible given the context. Thus, the correct inequality is 3p > 40, indicating the total food is indeed more than three times the weekly requirement.
The daily cost, C(x), for a company to produce x microscopes is given by the equation C(x) = 300 + 10.5x. What is the cost of producing 50 microscopes?
- A. $41,250
- B. $360.50
- C. $15,525
- D. $825
Correct Answer & Rationale
Correct Answer: D
To determine the cost of producing 50 microscopes, substitute x = 50 into the equation C(x) = 300 + 10.5x. This gives C(50) = 300 + 10.5(50) = 300 + 525 = 825. Thus, the total cost is $825. Option A ($41,250) is incorrect as it miscalculates the cost by multiplying incorrectly. Option B ($360.50) results from a misunderstanding of the equation, possibly neglecting the fixed cost. Option C ($15,525) likely arises from an error in multiplying the variable cost without adding the fixed cost. Each incorrect option fails to follow the proper calculation method outlined in the cost equation.
To determine the cost of producing 50 microscopes, substitute x = 50 into the equation C(x) = 300 + 10.5x. This gives C(50) = 300 + 10.5(50) = 300 + 525 = 825. Thus, the total cost is $825. Option A ($41,250) is incorrect as it miscalculates the cost by multiplying incorrectly. Option B ($360.50) results from a misunderstanding of the equation, possibly neglecting the fixed cost. Option C ($15,525) likely arises from an error in multiplying the variable cost without adding the fixed cost. Each incorrect option fails to follow the proper calculation method outlined in the cost equation.