Which graph represents the equation x - 2y = 4?
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A.
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B.
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C.
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D.
Correct Answer & Rationale
Correct Answer: A
To determine which graph represents the equation \( x - 2y = 4 \), we can rearrange it into slope-intercept form: \( y = \frac{1}{2}x - 2 \). This indicates a slope of \( \frac{1}{2} \) and a y-intercept at \( -2 \). Option A accurately reflects these characteristics, showing a line that rises gradually and crosses the y-axis at \( -2 \). Options B, C, and D do not have the correct slope or y-intercept. B has a steeper slope, C slopes downward, and D does not intersect the y-axis at the correct point. Thus, only Option A is consistent with the equation's graph.
To determine which graph represents the equation \( x - 2y = 4 \), we can rearrange it into slope-intercept form: \( y = \frac{1}{2}x - 2 \). This indicates a slope of \( \frac{1}{2} \) and a y-intercept at \( -2 \). Option A accurately reflects these characteristics, showing a line that rises gradually and crosses the y-axis at \( -2 \). Options B, C, and D do not have the correct slope or y-intercept. B has a steeper slope, C slopes downward, and D does not intersect the y-axis at the correct point. Thus, only Option A is consistent with the equation's graph.
Other Related Questions
Factor the expression completely: -3x - 21
- A. -3(x+7)
- B. -3(x-21)
- C. -3(x-7)
- D. -3(x+21)
Correct Answer & Rationale
Correct Answer: A
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
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 \(x = 2\).
Two men are employed at a local supermarket. The table shows James's earnings, and the graph shows Eric's earnings.
Based on the information above, who earns the greater amount per hour, and how much does he earn for a 7-hour shift?
- A. James earns the greater amount per hour and earns $73.50 for a 7-hour shift.
- B. James earns the greater amount per hour and earns $70.00 for a 7-hour shift.
- C. Eric earns the greater amount per hour and earns $70.00 for a 7-hour shift.
- D. Eric earns the greater amount per hour and earns $73.50 for a 7-hour shift.
Correct Answer & Rationale
Correct Answer: D
To determine who earns more per hour, one must compare the hourly rates of James and Eric. If Eric's hourly rate is higher, he earns more for a 7-hour shift, calculated as his hourly rate multiplied by 7. Option A incorrectly states James earns more and miscalculates his earnings. Option B also claims James earns more but provides the wrong total for a 7-hour shift. Option C correctly identifies Eric as the higher earner but misstates his total earnings for a 7-hour shift. Option D accurately identifies Eric as the higher earner and correctly calculates his earnings for a 7-hour shift at $73.50.
To determine who earns more per hour, one must compare the hourly rates of James and Eric. If Eric's hourly rate is higher, he earns more for a 7-hour shift, calculated as his hourly rate multiplied by 7. Option A incorrectly states James earns more and miscalculates his earnings. Option B also claims James earns more but provides the wrong total for a 7-hour shift. Option C correctly identifies Eric as the higher earner but misstates his total earnings for a 7-hour shift. Option D accurately identifies Eric as the higher earner and correctly calculates his earnings for a 7-hour shift at $73.50.
A figure is formed by shaded squares on a grid. Which figure has a perimeter of 12units and an area of 8 square units?
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A.
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B.
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C.
- D. None of the above
Correct Answer & Rationale
Correct Answer: C
To determine the figure that meets the criteria of having a perimeter of 12 units and an area of 8 square units, we analyze each option. Option C achieves both requirements: it has a perimeter of 12 units, calculated by adding the lengths of all sides, and an area of 8 square units, determined by multiplying its length and width (2 x 4). In contrast, Option A has a perimeter exceeding 12 units, while its area is less than 8 square units. Option B has a perimeter of 10 units and an area of 6 square units, failing both criteria. Option D is not applicable since Option C meets the conditions. Thus, Option C stands out as the only figure that satisfies both the perimeter and area requirements.
To determine the figure that meets the criteria of having a perimeter of 12 units and an area of 8 square units, we analyze each option. Option C achieves both requirements: it has a perimeter of 12 units, calculated by adding the lengths of all sides, and an area of 8 square units, determined by multiplying its length and width (2 x 4). In contrast, Option A has a perimeter exceeding 12 units, while its area is less than 8 square units. Option B has a perimeter of 10 units and an area of 6 square units, failing both criteria. Option D is not applicable since Option C meets the conditions. Thus, Option C stands out as the only figure that satisfies both the perimeter and area requirements.