The system of equations above has how many solutions? x+4y=3, 2x+8y=4
- A. None
- B. One
- C. Two
- D. Infinitely many
Correct Answer & Rationale
Correct Answer: A
To determine the number of solutions for the system of equations, we first analyze the equations: \(x + 4y = 3\) and \(2x + 8y = 4\). The second equation can be simplified by dividing all terms by 2, resulting in \(x + 4y = 2\). Now, we have two equations: \(x + 4y = 3\) and \(x + 4y = 2\). Since both equations represent parallel lines (same slope, different y-intercepts), they will never intersect, indicating there are no solutions. Option B suggests one solution, which is incorrect as parallel lines do not meet. Option C suggests two solutions, which is also incorrect for the same reason. Option D proposes infinitely many solutions, which applies only to identical lines, not parallel ones. Thus, the system has no solutions.
To determine the number of solutions for the system of equations, we first analyze the equations: \(x + 4y = 3\) and \(2x + 8y = 4\). The second equation can be simplified by dividing all terms by 2, resulting in \(x + 4y = 2\). Now, we have two equations: \(x + 4y = 3\) and \(x + 4y = 2\). Since both equations represent parallel lines (same slope, different y-intercepts), they will never intersect, indicating there are no solutions. Option B suggests one solution, which is incorrect as parallel lines do not meet. Option C suggests two solutions, which is also incorrect for the same reason. Option D proposes infinitely many solutions, which applies only to identical lines, not parallel ones. Thus, the system has no solutions.
Other Related Questions
If the trend shown in the graph above continued into the next year, approximately how many sport utility vehicles were sold in 1999?
- A. 3 million
- B. 2.5 million
- C. 2 million
- D. 3 thousand
Correct Answer & Rationale
Correct Answer: A
To determine the approximate number of sport utility vehicles sold in 1999, analyzing the trend in the graph is essential. If the upward trend continued, sales would likely increase compared to previous years. Given the data, 3 million aligns with the projected growth rate, reflecting a significant rise consistent with market trends. Option B, 2.5 million, underestimates the growth, while C, 2 million, does not account for the upward trajectory. Option D, 3 thousand, is far too low and unrealistic, failing to represent the scale of SUV sales during that period. Thus, 3 million is the most reasonable estimate.
To determine the approximate number of sport utility vehicles sold in 1999, analyzing the trend in the graph is essential. If the upward trend continued, sales would likely increase compared to previous years. Given the data, 3 million aligns with the projected growth rate, reflecting a significant rise consistent with market trends. Option B, 2.5 million, underestimates the growth, while C, 2 million, does not account for the upward trajectory. Option D, 3 thousand, is far too low and unrealistic, failing to represent the scale of SUV sales during that period. Thus, 3 million is the most reasonable estimate.
How many cups of peanut butter must be used in order to make exactly enough peanut butter balls for the children at the party?
- A. 10
- B. 12
- C. 18
- D. 24
Correct Answer & Rationale
Correct Answer: C
To determine the number of cups of peanut butter needed for the peanut butter balls, one must consider the recipe's requirements and the number of children attending the party. Option C (18 cups) aligns with the recipe's proportion to yield the exact quantity necessary for all children. Option A (10 cups) is insufficient, likely resulting in fewer peanut butter balls than required. Option B (12 cups) may also fall short, leading to a shortage. Option D (24 cups) exceeds the needed amount, creating waste. Thus, C is the optimal choice, ensuring each child receives a peanut butter ball without excess or deficit.
To determine the number of cups of peanut butter needed for the peanut butter balls, one must consider the recipe's requirements and the number of children attending the party. Option C (18 cups) aligns with the recipe's proportion to yield the exact quantity necessary for all children. Option A (10 cups) is insufficient, likely resulting in fewer peanut butter balls than required. Option B (12 cups) may also fall short, leading to a shortage. Option D (24 cups) exceeds the needed amount, creating waste. Thus, C is the optimal choice, ensuring each child receives a peanut butter ball without excess or deficit.
If a +√x= b then x =
- A. √b-√a
- B. √(b-1)
- C. (b-a)²
- D. b²-a²
Correct Answer & Rationale
Correct Answer: C
To solve for \( x \) in the equation \( a + \sqrt{x} = b \), we first isolate \( \sqrt{x} \) by rearranging the equation to \( \sqrt{x} = b - a \). Squaring both sides gives \( x = (b - a)^2 \), which corresponds to option C. Option A, \( \sqrt{b} - \sqrt{a} \), does not account for squaring the expression and thus cannot represent \( x \). Option B, \( \sqrt{(b-1)} \), is unrelated to the original equation and lacks the necessary operations. Option D, \( b^2 - a^2 \), applies the difference of squares incorrectly and does not solve for \( x \) directly.
To solve for \( x \) in the equation \( a + \sqrt{x} = b \), we first isolate \( \sqrt{x} \) by rearranging the equation to \( \sqrt{x} = b - a \). Squaring both sides gives \( x = (b - a)^2 \), which corresponds to option C. Option A, \( \sqrt{b} - \sqrt{a} \), does not account for squaring the expression and thus cannot represent \( x \). Option B, \( \sqrt{(b-1)} \), is unrelated to the original equation and lacks the necessary operations. Option D, \( b^2 - a^2 \), applies the difference of squares incorrectly and does not solve for \( x \) directly.
If a number from set M is selected at random, what is the probability that the number selected will be a factor of 12?
- A. 0.1
- B. 0.2
- C. 0.4
- D. 0.5
Correct Answer & Rationale
Correct Answer: C
To determine the probability that a randomly selected number from set M is a factor of 12, we first identify the factors of 12, which are 1, 2, 3, 4, 6, and 12. If set M consists of 6 numbers (1 through 6), then 4 of these (1, 2, 3, and 4) are factors of 12. Thus, the probability is 4 out of 6, simplifying to 0.4. Option A (0.1) underestimates the number of factors. Option B (0.2) suggests only 2 factors, which is incorrect. Option D (0.5) implies 3 factors, also inaccurate. Therefore, 0.4 accurately represents the proportion of factors of 12 in the set.
To determine the probability that a randomly selected number from set M is a factor of 12, we first identify the factors of 12, which are 1, 2, 3, 4, 6, and 12. If set M consists of 6 numbers (1 through 6), then 4 of these (1, 2, 3, and 4) are factors of 12. Thus, the probability is 4 out of 6, simplifying to 0.4. Option A (0.1) underestimates the number of factors. Option B (0.2) suggests only 2 factors, which is incorrect. Option D (0.5) implies 3 factors, also inaccurate. Therefore, 0.4 accurately represents the proportion of factors of 12 in the set.