Let's go through each question one by one:
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How many multiple-choice questions are on the test? Given the system of equations: \[ m + f = 15 \] \[ 3m + 5f = 51 \] We can solve this system. From the first equation, we can express \(f\) in terms of \(m\): \[ f = 15 - m \] Substituting \(f\) into the second equation: \[ 3m + 5(15 - m) = 51 \] This simplifies to: \[ 3m + 75 - 5m = 51 \] \[ -2m + 75 = 51 \] \[ -2m = 51 - 75 \] \[ -2m = -24 \] \[ m = 12 \] Thus, there are 12 multiple-choice questions.
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Which system of linear equations can be used for Yolanda’s coins? The total number of coins is 28 and they total $4: \[ n + q = 28 \] \[ 0.05n + 0.25q = 4 \] So the correct system is: \[ n + q = 28 \quad 0.05n + 0.25q = 4 \]
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What was the speed of the wind? Let \(p\) be the speed of the plane and \(w\) be the wind speed. The two equations based on distance = rate × time are:
- Into the wind: \((p - w) \cdot 2.5 = 400\) → \(p - w = 160\) (1).
- With the wind: \((p + w) \cdot 2 = 400\) → \(p + w = 200\) (2). Adding equations (1) and (2): \[ (p - w) + (p + w) = 160 + 200 \] \[ 2p = 360 \Rightarrow p = 180 \] Substituting \(p\) back into (1): \[ 180 - w = 160 \Rightarrow w = 20 \] Thus, the speed of the wind is 20 mph.
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Which system of equations can be used for the motorboat? The time taken downstream is 30 minutes (0.5 hours), and the time taken upstream is 90 minutes (1.5 hours). The equations based on distance = rate × time are: \[ 9 = 0.5(x + y) \quad \text{(downstream)} \] \[ 9 = 1.5(x - y) \quad \text{(upstream)} \] Therefore, the correct system is: \[ 9 = 0.5(x + y) \quad 9 = 1.5(x - y) \]
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What is the value of the nickels Bethany has? Let \(n\) be nickels and \(d\) be dimes. We have: \[ n + d = 44 \] \[ 0.05n + 0.10d = 3.10 \] Solving these equations: From the first, \(d = 44 - n\). Substitute into the second: \[ 0.05n + 0.10(44 - n) = 3.10 \] \[ 0.05n + 4.4 - 0.10n = 3.10 \] \[ -0.05n = -1.3 \Rightarrow n = 26 \] Nickels' value: \(0.05 \times 26 = 1.30\). So, the value of the nickels is $1.30.
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What is the speed of the current for Micah? Let \(b\) be the speed of the boat and \(c\) be the speed of the current: Downstream: \[ 9 = 0.32(b + c) \Rightarrow 28.125 = b + c \] Upstream: \[ 9 = 0.56(b - c) \Rightarrow 16.0714 = b - c \] Solving this system gives you the current speed 3 mph.
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Which system of linear equations represents the aircraft situation? Using the distance formula: \[ 0.75(x + y) = 120 \] \[ 1(x - y) = 120 \] The correct system is: \[ 0.75(x + y) = 120 \quad 1(x - y) = 120 \]
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Which system of equations for Lily's sales? Let \(b\) be bracelets and \(n\) necklaces: \[ b + n = 18 \] \[ 6b + 5n = 101 \] The correct system is: \[ b + n = 18 \quad 6b + 5n = 101 \]
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How much does a salesperson earn for certain sales? Lead to: \[ c = 800 + 200t \] Thus, the earned for selling \(2\) cars \(t=6\), and \(3\) trucks leads to: The answer reduces to $2,700.
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What does x represent for Jeremiah’s job? The variable \(x\) signifies the number of hours worked per week.
Feel free to ask if you need further clarifications or details on any specific problem!