Which graph shows 3y - 2x = 6?
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A.
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B.
Correct Answer & Rationale
Correct Answer: B
To determine the graph of the equation \(3y - 2x = 6\), we can rearrange it into slope-intercept form \(y = mx + b\). This gives us \(y = \frac{2}{3}x + 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of 2. Option B accurately represents this line, showing the correct slope and intercept. In contrast, Option A does not align with the expected slope or y-intercept, thus failing to represent the equation correctly. The visual representation in Option B confirms the relationship defined by the equation.
To determine the graph of the equation \(3y - 2x = 6\), we can rearrange it into slope-intercept form \(y = mx + b\). This gives us \(y = \frac{2}{3}x + 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of 2. Option B accurately represents this line, showing the correct slope and intercept. In contrast, Option A does not align with the expected slope or y-intercept, thus failing to represent the equation correctly. The visual representation in Option B confirms the relationship defined by the equation.
Other Related Questions
A diver jumps from a platform. The height, h meters, the diver is above the water t seconds after jumping is represented by h = -16t^2 + 16t + 6.5. To the near hundredth of a second, how many seconds after jumping is the diver 2.5 meters above the water?
- A. 2.79
- B. 1.32
- C. 2.83
- D. 1.21
Correct Answer & Rationale
Correct Answer: D
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
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.
What is the value of 2/5 multiplied by 5/4 divided by 4/3
- A. 32/75
- B. 3\8
- C. 6\25
- D. 2\3
Correct Answer & Rationale
Correct Answer: B
To solve \( \frac{2}{5} \times \frac{5}{4} \div \frac{4}{3} \), we first multiply \( \frac{2}{5} \) by \( \frac{5}{4} \). This results in \( \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2} \). Next, dividing by \( \frac{4}{3} \) is the same as multiplying by its reciprocal, \( \frac{3}{4} \). Therefore, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Option A, \( \frac{32}{75} \), is incorrect as it does not simplify from the given operations. Option C, \( \frac{6}{25} \), results from miscalculating the division. Option D, \( \frac{2}{3} \), is also incorrect as it doesn't follow from the correct operations.
To solve \( \frac{2}{5} \times \frac{5}{4} \div \frac{4}{3} \), we first multiply \( \frac{2}{5} \) by \( \frac{5}{4} \). This results in \( \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2} \). Next, dividing by \( \frac{4}{3} \) is the same as multiplying by its reciprocal, \( \frac{3}{4} \). Therefore, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Option A, \( \frac{32}{75} \), is incorrect as it does not simplify from the given operations. Option C, \( \frac{6}{25} \), results from miscalculating the division. Option D, \( \frac{2}{3} \), is also incorrect as it doesn't follow from the correct operations.
Select the factors for the following expression 2x^2 - xy - 3y^2
- A. (2x+3y)(x-y)
- B. (x+y)(2x-3y)
- C. (2x-y)(x+3y)
- D. (2x-3y)(x+y)
Correct Answer & Rationale
Correct Answer: D
To factor the expression \(2x^2 - xy - 3y^2\), we look for two binomials that multiply to give the original expression. Option D, \((2x-3y)(x+y)\), expands to \(2x^2 + 2xy - 3xy - 3y^2\), which simplifies to \(2x^2 - xy - 3y^2\), matching the original expression. Option A, \((2x+3y)(x-y)\), expands to \(2x^2 - 2xy + 3xy - 3y^2\), resulting in \(2x^2 + xy - 3y^2\), which is incorrect. Option B, \((x+y)(2x-3y)\), gives \(2x^2 - 3xy + 2xy - 3y^2\), simplifying to \(2x^2 - xy - 3y^2\), but the signs do not match the original expression. Option C, \((2x-y)(x+3y)\), expands to \(2x^2 + 6xy - xy - 3y^2\), leading to \(2x^2 + 5xy - 3y^2\), which is also incorrect. Thus, only Option D correctly factors the expression.
To factor the expression \(2x^2 - xy - 3y^2\), we look for two binomials that multiply to give the original expression. Option D, \((2x-3y)(x+y)\), expands to \(2x^2 + 2xy - 3xy - 3y^2\), which simplifies to \(2x^2 - xy - 3y^2\), matching the original expression. Option A, \((2x+3y)(x-y)\), expands to \(2x^2 - 2xy + 3xy - 3y^2\), resulting in \(2x^2 + xy - 3y^2\), which is incorrect. Option B, \((x+y)(2x-3y)\), gives \(2x^2 - 3xy + 2xy - 3y^2\), simplifying to \(2x^2 - xy - 3y^2\), but the signs do not match the original expression. Option C, \((2x-y)(x+3y)\), expands to \(2x^2 + 6xy - xy - 3y^2\), leading to \(2x^2 + 5xy - 3y^2\), which is also incorrect. Thus, only Option D correctly factors the expression.