Evaluating Expressions - SHSAT Math

1. When $x = 2$ and $y = -1$, what is the value of $\frac{x^3 + 2xy - y^2}{x + 3y}$?
   Answer: $-3$ Substitute $x = 2$ and $y = -1$: First calculate the numerator: $x^3 + 2xy - y^2 = 2^3 + 2(2)(-1) - (-1)^2 = 8 + (-4) - 1 = 3$. Then calculate the denominator: $x + 3y = 2 + 3(-1) = 2 - 3 = -1$. Therefore: $\frac{x^3 + 2xy - y^2}{x + 3y} = \frac{3}{-1} = -3$. Choice B (-1) results from using just the denominator value. Choice C (5) comes from calculating the numerator as $8 - 4 + 1 = 5$ (sign error with $y^2$) and ignoring the negative denominator. Choice D (7) results from calculating the numerator as $8 + 4 - 1 = 11$ (sign error with $2xy$) but making additional errors.
2. What is the value of $5x-3$ when $x=-2$ ?
   Answer: −13 This question tests substitution, a fundamental algebra skill where you replace a variable with a given value and simplify the expression. When you see $x = -2$, you need to substitute $-2$ for every $x$ in the expression $5x - 3$. This gives you $5(-2) - 3$. Following order of operations, multiply first: $5(-2) = -10$. Then subtract: $-10 - 3 = -13$. Let's examine why each answer choice might appear: A) $-13$ is correct, as shown above. B) $-7$ likely comes from incorrectly handling the negative signs. You might get this if you calculated $5(-2) = -10$ correctly but then computed $-10 - 3$ as $-7$ instead of $-13$. Remember that subtracting a positive number from a negative number makes the result more negative. C) $2$ suggests a sign error early in the calculation. This could happen if you treated $5(-2)$ as $+10$ instead of $-10$, then calculated $10 - 3 = 7$, and made another error to reach $2$. D) $13$ is the positive version of the correct answer. This occurs when you get the right magnitude but wrong sign, possibly from calculating $5(2) - 3$ instead of $5(-2) - 3$, or from making multiple sign errors that coincidentally produce the right absolute value. Study tip: When substituting negative values, put parentheses around them to avoid sign errors. Write $5(-2) - 3$ rather than $5-2-3$ to keep your work clear and accurate.
3. When $a = 4$ and $b = -2$, what is the value of $3a^2 - 2ab + b^2 - 5a$?
   Answer: $48$ Substitute $a = 4$ and $b = -2$ and evaluate each term separately, watching the signs. $3a^2 = 3(16) = 48$ $-2ab = -2(4)(-2) = +16$ $b^2 = (-2)^2 = 4$ $-5a = -5(4) = -20$ Add: $48 + 16 + 4 - 20 = 48$. The value 24 comes from treating $a^2$ as $2a$, so the first term becomes 24 instead of 48. The value 36 comes from a sign slip on $-2ab$ (using $-16$) and then leaving off the $-5a$ term. Remember that a negative times a negative is positive, and that squaring $-2$ gives $+4$.
4. If $a = -2$, $b = 3$, and $c = 1$, what is the value of $a^2b - 3abc + c^3$?
   Answer: $31$ Substitute $a = -2$, $b = 3$, and $c = 1$: $a^2b - 3abc + c^3 = (-2)^2 \cdot 3 - 3(-2)(3)(1) + 1^3 = 4 \cdot 3 - 3(-2)(3)(1) + 1 = 12 - 3(-6) + 1 = 12 + 18 + 1 = 31$. Choice A (-5) results from calculating $a^2 = 4$, $a^2b = 12$, $3abc = 3(-2)(3)(1) = -18$, but then computing $12 - (-18) + 1$ incorrectly as $12 - 18 + 1 = -5$. Choice B (7) comes from sign errors in multiple terms. Choice C (13) results from calculating $3abc = 18$ instead of $-18$, giving $12 - 18 + 1 = -5$, then making additional sign errors.

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