Simplify: (3x - 5) + (-7x + 2)
- A. -4x^2 - 3
- B. -4x - 3
- C. 28
- D. -4x^2 - 10
Correct Answer & Rationale
Correct Answer: B
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
Other Related Questions
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.
How many more miles did the space shuttle Discovery travel than the space shuttle Atlantis?
- A. 274,100,000 miles
- B. 274,100 miles
- C. 22.3 miles
- D. 22,300,000 miles
Correct Answer & Rationale
Correct Answer: D
To determine the difference in miles traveled between the space shuttles Discovery and Atlantis, one must subtract the total miles of Atlantis from Discovery. The calculation reveals that Discovery traveled 22,300,000 miles more than Atlantis, making option D the accurate choice. Option A, 274,100,000 miles, is excessively high and does not reflect the actual difference. Option B, 274,100 miles, is too low and misrepresents the scale of space travel. Option C, 22.3 miles, is trivial and fails to capture the vast distances involved in space missions. Thus, option D accurately represents the significant difference in miles traveled.
To determine the difference in miles traveled between the space shuttles Discovery and Atlantis, one must subtract the total miles of Atlantis from Discovery. The calculation reveals that Discovery traveled 22,300,000 miles more than Atlantis, making option D the accurate choice. Option A, 274,100,000 miles, is excessively high and does not reflect the actual difference. Option B, 274,100 miles, is too low and misrepresents the scale of space travel. Option C, 22.3 miles, is trivial and fails to capture the vast distances involved in space missions. Thus, option D accurately represents the significant difference in miles traveled.
Which equation represents the graphed line?
- A. y = -1/3x +3
- B. y = 3x - 7
- C. y = 3x + 7
- D. y = 1/3x + 1
Correct Answer & Rationale
Correct Answer: D
The equation y = 1/3x + 1 accurately represents the graphed line due to its positive slope of 1/3, indicating a gradual upward rise, consistent with the line’s direction. The y-intercept of 1 shows that the line crosses the y-axis at the point (0, 1), aligning perfectly with the graph. Option A, with a slope of -1/3, suggests a downward trend, which contradicts the graph’s upward slope. Option B has a much steeper slope of 3, leading to a different angle of rise. Option C also has a slope of 3 and a y-intercept of 7, which does not match the graph’s intercept. Thus, only D accurately reflects both the slope and intercept of the line shown.
The equation y = 1/3x + 1 accurately represents the graphed line due to its positive slope of 1/3, indicating a gradual upward rise, consistent with the line’s direction. The y-intercept of 1 shows that the line crosses the y-axis at the point (0, 1), aligning perfectly with the graph. Option A, with a slope of -1/3, suggests a downward trend, which contradicts the graph’s upward slope. Option B has a much steeper slope of 3, leading to a different angle of rise. Option C also has a slope of 3 and a y-intercept of 7, which does not match the graph’s intercept. Thus, only D accurately reflects both the slope and intercept of the line shown.
At a local bank, certificates of deposit (CDs) mature every 9 months. At another bank, CDs mature every 12 months. If CDs are purchased on the same day at each bank and are renewed when they mature, what is the least number of months that will pass before the two banks' CDs are mature at the same time?
- A. 72
- B. 36
- C. 108
- D. 3
Correct Answer & Rationale
Correct Answer: B
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.