Simplify and evaluate algebraic expressions - TACHS Math

1. For $t=-2$, find $\dfrac{t^{2}-4t}{t}$.
   Answer: $-6$ This problem tests algebraic substitution and simplification with negative numbers. When you see an algebraic expression to evaluate at a specific value, substitute carefully and follow the order of operations. Let's substitute $t = -2$ into $\frac{t^2 - 4t}{t}$: $\frac{(-2)^2 - 4(-2)}{-2}$ First, evaluate the numerator: $(-2)^2 = 4$ and $4(-2) = -8$, so the numerator becomes $4 - (-8) = 4 + 8 = 12$. Now we have $\frac{12}{-2} = -6$. Answer choice A ($12$) represents the value of just the numerator, forgetting to divide by $t = -2$. This is a common error when students calculate the pieces correctly but don't complete the final step. Answer choice B ($6$) comes from correctly finding the numerator as $12$ but then dividing by $2$ instead of $-2$, losing track of the negative sign. Answer choice C ($-12$) results from getting the wrong sign in the numerator calculation, likely computing $4 - 4(-2) = 4 - 8 = -4$ incorrectly, then getting $\frac{-4}{-2} = 2$ and making additional errors, or from $\frac{-12}{-2}$ if the numerator was miscalculated as $-12$. Answer choice D ($-6$) is correct. When substituting negative values, write them in parentheses to avoid sign errors. Also, consider simplifying algebraically first: $\frac{t^2 - 4t}{t} = \frac{t(t-4)}{t} = t-4$, then substitute to get $-2-4 = -6$.
2. Simplify $4b^{2} + 3b - 2b^{2} - 5b + 1$.
   Answer: $2b^{2} - 2b + 1$ When you see an expression with multiple terms containing the same variable, your goal is to combine like terms by grouping terms with identical variable parts and adding their coefficients. To simplify $4b^{2} + 3b - 2b^{2} - 5b + 1$, first identify and group the like terms. You have three types: $b^{2}$ terms, $b$ terms, and constant terms. Rearranging: $(4b^{2} - 2b^{2}) + (3b - 5b) + 1$. Now combine each group by adding the coefficients. For the $b^{2}$ terms: $4 - 2 = 2$, so you get $2b^{2}$. For the $b$ terms: $3 - 5 = -2$, giving you $-2b$. The constant remains $1$. Your final answer is $2b^{2} - 2b + 1$. Choice A ($6b^{2} - 8b + 1$) incorrectly adds the $b^{2}$ coefficients as $4 + 2 = 6$ instead of $4 - 2 = 2$, and adds the $b$ coefficients as $3 + 5 = 8$ instead of $3 - 5 = -2$. Choice C ($2b^{2} + 8b - 1$) correctly finds the $b^{2}$ term but makes the same sign errors as choice A for the $b$ term and constant. Choice D ($-2b^{2} - 2b + 1$) gets the $b$ term and constant correct but incorrectly calculates the $b^{2}$ coefficient as $-2$ instead of $+2$. Always pay careful attention to signs when combining like terms. Write out the operation explicitly (like $4 - 2$) rather than doing it mentally to avoid sign errors.
3. Let $m = \dfrac{1}{2}$. What is $8m^{2} - 3m + 2$?
   Answer: $\dfrac{5}{2}$ When you encounter an expression with a given variable value, you need to substitute carefully and follow the order of operations. This tests your ability to work with fractions and evaluate algebraic expressions systematically. Let's substitute $m = \frac{1}{2}$ into $8m^2 - 3m + 2$. First, calculate $m^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$. Now substitute both values: $8m^2 - 3m + 2 = 8 \cdot \frac{1}{4} - 3 \cdot \frac{1}{2} + 2$ Simplify each term: $8 \cdot \frac{1}{4} = 2$, $3 \cdot \frac{1}{2} = \frac{3}{2}$, and the constant remains $2$. So we have: $2 - \frac{3}{2} + 2 = 4 - \frac{3}{2}$ Convert to a common denominator: $4 = \frac{8}{2}$, so $\frac{8}{2} - \frac{3}{2} = \frac{5}{2}$. The answer is C. Let's examine the wrong choices: Choice A ($\frac{1}{2}$) likely results from only calculating one term or making multiple arithmetic errors. Choice B ($\frac{7}{2}$) could come from incorrectly adding instead of subtracting the middle term, giving $4 + \frac{3}{2} = \frac{11}{2}$, then making another error. Choice D ($\frac{11}{4}$) might result from calculation errors with the fractions or incorrect order of operations. Always substitute step-by-step, square fractions carefully (remember that $\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}$), and convert to common denominators before adding or subtracting fractions. Double-check your arithmetic with fraction operations.
4. Simplify $-3(2k - 5) + 4(k + 3)$.
   Answer: $-2k + 27$ When you encounter an expression with multiple terms in parentheses, you need to apply the distributive property carefully to each part, then combine like terms. Start by distributing each coefficient to the terms inside its parentheses. For $-3(2k - 5)$, multiply $-3$ by both $2k$ and $-5$: $-3 \cdot 2k = -6k$ and $-3 \cdot (-5) = +15$. This gives you $-6k + 15$. For $4(k + 3)$, multiply $4$ by both $k$ and $3$: $4 \cdot k = 4k$ and $4 \cdot 3 = 12$. This gives you $4k + 12$. Now your expression becomes: $-6k + 15 + 4k + 12$. Combine like terms by grouping the $k$ terms and the constants: $(-6k + 4k) + (15 + 12) = -2k + 27$. Looking at the wrong answers: Choice A ($-2k - 27$) correctly finds the $k$ coefficient but makes a sign error with the constants—likely from miscalculating $15 + 12$ or the distribution. Choice B ($2k + 27$) gets the constant term right but makes a sign error with the $k$ coefficient, probably from $-6 + 4 = 2$ instead of $-2$. Choice D ($2k - 27$) combines both sign errors from choices A and B. The key strategy here is to work methodically: distribute first, then combine like terms. Double-check your signs at each step, especially when distributing negative coefficients—this is where most errors occur on these problems.
5. Simplify $\dfrac{1}{3}(9x - 6) + 4$.
   Answer: $3x + 2$ This problem tests your ability to use the distributive property and combine like terms—two fundamental algebraic skills that appear frequently on standardized tests. To simplify $\frac{1}{3}(9x - 6) + 4$, start by distributing $\frac{1}{3}$ to both terms inside the parentheses. Multiply $\frac{1}{3} \times 9x = 3x$ and $\frac{1}{3} \times (-6) = -2$. This gives you $3x - 2 + 4$. Finally, combine the constant terms: $-2 + 4 = 2$. The simplified expression is $3x + 2$, which is answer choice C. Let's examine why the other options are incorrect. Choice A, $\frac{3x}{2} + 4$, results from incorrectly distributing $\frac{1}{3}$ to only the $9x$ term, giving $\frac{9x}{3} = 3x$ but then mistakenly writing it as $\frac{3x}{2}$, and failing to distribute to the $-6$ at all. Choice B, $3x - 2$, correctly distributes $\frac{1}{3}$ to get $3x - 2$, but then forgets to add the $+4$ term at the end. Choice D, $6x + 4$, appears to result from multiplying $9x$ by $\frac{2}{3}$ instead of $\frac{1}{3}$, or from some other fundamental error in distribution. When simplifying expressions with distribution, work systematically: distribute first, then combine like terms. Double-check that you've applied the distributive property to every term inside the parentheses, and don't forget about terms outside the parentheses that need to be combined at the end.
6. Simplify the expression $3x + 4x - 2 + 5$.
   Answer: $7x + 3$ When you see an algebraic expression with multiple terms, your goal is to combine like terms—terms that have the same variable part or are both constants. Let's work through $3x + 4x - 2 + 5$ step by step. First, identify the like terms: $3x$ and $4x$ are like terms (both have the variable $x$), while $-2$ and $+5$ are like terms (both are constants with no variables). Combine the $x$ terms: $3x + 4x = 7x$. Think of this as having 3 of something plus 4 more of the same thing, giving you 7 total. Combine the constant terms: $-2 + 5 = 3$. You're starting at -2 on a number line and moving 5 positions to the right, landing at 3. Your final simplified expression is $7x + 3$, which is answer choice C. Now let's see where the wrong answers come from. Choice A ($-7x + 3$) incorrectly makes the coefficient of $x$ negative—this happens if you mistakenly subtract instead of adding the $x$ terms. Choice B ($7x - 3$) gets the $x$ term right but makes the constant negative, likely from calculating $-2 + 5$ as $-3$ instead of $+3$. Choice D ($-7x - 3$) combines both errors, getting both the coefficient and constant wrong. Remember: when combining like terms, pay careful attention to the signs in front of each term. It helps to rewrite subtraction as adding a negative: $3x + 4x + (-2) + 5$.
7. Evaluate $2a^{2} - 3a + 4$ when $a = -2$.
   Answer: $18$ When you see an algebraic expression that needs to be evaluated at a specific value, you're being tested on substitution and order of operations. The key is to carefully replace the variable with the given value and follow PEMDAS. To evaluate $2a^{2} - 3a + 4$ when $a = -2$, substitute $-2$ for every $a$: $2(-2)^{2} - 3(-2) + 4$ Now follow the order of operations. First, handle the exponent: $(-2)^{2} = 4$ (remember that a negative number squared becomes positive). This gives you: $2(4) - 3(-2) + 4$ Next, perform the multiplications: $2(4) = 8$ and $-3(-2) = 6$ (negative times negative equals positive). You now have: $8 + 6 + 4$ Finally, add from left to right: $8 + 6 + 4 = 18$ Looking at the wrong answers: Choice A ($-2$) likely comes from incorrectly calculating $(-2)^{2} = -4$ instead of $4$. Choice B ($-18$) probably results from sign errors, such as getting $-3(-2) = -6$ instead of $6$. Choice D ($0$) might occur if you make multiple computational mistakes that coincidentally cancel out. The most common trap here is mishandling negative numbers, especially with exponents and multiplication. Remember: $(-2)^{2} = 4$, but $-2^{2} = -4$. Always use parentheses when substituting negative values, and double-check your signs at each step.
8. Which expression is equivalent to $2(x + 4) - (3x - 5)$?
   Answer: $-x + 13$ When you see an expression with parentheses and subtraction like this, you need to carefully distribute and combine like terms. The key is handling the negative sign in front of the second set of parentheses correctly. Let's work through $2(x + 4) - (3x - 5)$ step by step. First, distribute the 2 into the first parentheses: $2(x + 4) = 2x + 8$. Next, handle the subtraction of the second expression. When you subtract $(3x - 5)$, you're distributing a negative sign to each term inside: $-(3x - 5) = -3x + 5$. Notice that subtracting a negative 5 becomes positive 5. Now combine: $2x + 8 - 3x + 5$. Group like terms: $(2x - 3x) + (8 + 5) = -x + 13$. Choice A ($5x - 13$) likely comes from incorrectly adding the x-coefficients (2 + 3 = 5) and subtracting the constants incorrectly. Choice C ($x - 13$) suggests adding the x-coefficients as 2 + 1 = 3, then somehow getting just x, while also getting the wrong constant. Choice D ($-x - 13$) correctly finds the x-term but makes the same constant error as choices A and C, probably from calculating $8 - 5 = 3$ instead of $8 + 5 = 13$. The correct answer is B: $-x + 13$. Study tip: When subtracting expressions in parentheses, rewrite the problem by distributing the negative sign first. This prevents sign errors that are extremely common on standardized tests.
9. Simplify: $7 - \bigl[2(3n - 4) - (n - 6)\bigr]$.
   Answer: $-5n + 9$ When you encounter expressions with nested brackets and parentheses, work systematically from the inside out, carefully tracking positive and negative signs throughout each step. Start by simplifying inside the brackets: $2(3n - 4) - (n - 6)$. Distribute the 2: $2(3n - 4) = 6n - 8$. For the second term, distribute the negative sign: $-(n - 6) = -n + 6$. Combining these gives: $6n - 8 - n + 6 = 5n - 2$. Now substitute back into the original expression: $7 - [5n - 2]$. Distribute the negative sign in front of the brackets: $7 - 5n + 2 = -5n + 9$. Choice A ($5n - 9$) results from forgetting to distribute the negative sign in front of the brackets, keeping $5n$ positive instead of making it $-5n$. Choice B ($5n + 9$) makes the same sign error with the $n$ term but also incorrectly handles the constant terms. Choice C ($-5n - 9$) correctly makes the $n$ term negative but fails to properly distribute the negative sign to the constant term $-2$, which should become $+2$. The correct answer is D: $-5n + 9$. Remember that distributing a negative sign changes the sign of every term inside the brackets or parentheses. When simplifying complex expressions, work methodically through one set of grouping symbols at a time, and double-check your sign changes at each step.
10. Simplify $\dfrac{5x - 10}{5} + \dfrac{3x}{5}$.
    Answer: $\dfrac{8x}{5} - 2$ When you see fractions with the same denominator, you can combine them by adding or subtracting the numerators while keeping the denominator the same. This problem tests your ability to distribute and combine like terms within fraction addition. Let's work through this step by step. First, notice that both fractions have the same denominator of 5, so we can combine them: $\dfrac{5x - 10}{5} + \dfrac{3x}{5} = \dfrac{(5x - 10) + 3x}{5}$ Now simplify the numerator by combining like terms: $(5x - 10) + 3x = 5x + 3x - 10 = 8x - 10$ So our expression becomes: $\dfrac{8x - 10}{5}$ We can split this into separate terms: $\dfrac{8x}{5} - \dfrac{10}{5} = \dfrac{8x}{5} - 2$ This matches answer choice C. Let's examine why the other choices are wrong. Choice A gives $\dfrac{2x}{5} - 2$, which incorrectly combines the x-terms as $5x + 3x = 2x$ instead of $8x$. Choice B shows $\dfrac{8x}{5} + 2$, which gets the x-term right but makes the constant positive instead of negative—forgetting that $-10 ÷ 5 = -2$. Choice D combines both errors: wrong x-coefficient and wrong sign on the constant. Remember: when adding fractions with the same denominator, combine the numerators carefully and watch your signs. Always double-check your arithmetic, especially when distributing negative signs and combining like terms.
11. If $x = 6$, what is the value of $\dfrac{3x}{2} - 4$?
    Answer: $5$ This question tests your ability to substitute values into algebraic expressions and follow the order of operations correctly. When you're given that $x = 6$, you need to substitute this value into the expression $\frac{3x}{2} - 4$ and calculate step by step. First, substitute: $\frac{3(6)}{2} - 4$. Next, multiply in the numerator: $\frac{18}{2} - 4$. Then divide: $9 - 4$. Finally, subtract: $5$. Let's examine why each wrong answer appears. Choice A ($-5$) likely comes from making a sign error somewhere in the calculation, perhaps treating the subtraction incorrectly or making an arithmetic mistake that flips the sign. Choice B ($13$) suggests you might have added 4 instead of subtracting it, getting $9 + 4 = 13$. Choice C ($\frac{7}{2}$) indicates you probably subtracted 4 from the numerator before dividing: $\frac{18-4}{2} = \frac{14}{2} = 7$, then mistakenly left it as $\frac{7}{2}$ instead of converting to the whole number 7. This violates the order of operations since you must complete the division $\frac{18}{2}$ before subtracting 4. The key strategy here is to always follow the order of operations (PEMDAS) carefully. When substituting values into expressions, work methodically through each step and double-check your arithmetic. Watch especially for the common trap of performing operations in the wrong sequence when fractions are involved.
12. Simplify $5\,(2y - 3) - 2\,(y - 1)$.
    Answer: $8y - 13$ This problem tests your ability to distribute multiplication over parentheses and combine like terms—two fundamental algebraic skills that often appear together. Start by distributing each coefficient to the terms inside the parentheses. For the first part, $5(2y - 3)$, multiply 5 by each term: $5 \times 2y = 10y$ and $5 \times (-3) = -15$. For the second part, $-2(y - 1)$, multiply -2 by each term: $-2 \times y = -2y$ and $-2 \times (-1) = +2$. This gives you $10y - 15 - 2y + 2$. Now combine like terms by grouping the $y$ terms and the constant terms separately: $(10y - 2y) + (-15 + 2) = 8y - 13$. This matches choice B. Choice A ($10y - 5$) represents the error of only distributing the first term and incorrectly combining constants. Choice C ($3y - 1$) occurs when you subtract the coefficients incorrectly ($10 - 2 = 3$ instead of $10 - 2 = 8$) and make sign errors with the constants. Choice D ($8y + 13$) gets the coefficient right but flips the sign on the constant term, likely from mishandling the negative distribution. When distributing negative signs, be extra careful with your arithmetic. Write out each step clearly: distribute first, then combine like terms. Double-check your signs—negative distributions are where most students make errors on problems like this.
13. Evaluate $\left(\dfrac{p-1}{3}\right)^{2}$ for $p = 7$.
    Answer: $4$ This question tests your ability to substitute values into algebraic expressions and follow the order of operations carefully. To evaluate $\left(\dfrac{p-1}{3}\right)^{2}$ when $p = 7$, start by substituting the given value: $\left(\dfrac{7-1}{3}\right)^{2}$. Next, simplify inside the parentheses first. Calculate $7-1 = 6$, giving you $\left(\dfrac{6}{3}\right)^{2}$. Then simplify the fraction: $\dfrac{6}{3} = 2$, so you have $(2)^{2}$. Finally, square the result: $2^{2} = 4$. The answer is D. Let's examine why the other choices are incorrect. Choice A ($\dfrac{64}{9}$) results from a common error where students square both the numerator and denominator separately without simplifying first, getting $\left(\dfrac{6}{3}\right)^{2} = \dfrac{6^{2}}{3^{2}} = \dfrac{36}{9} = 4$. Wait—that's still 4. Actually, choice A comes from squaring 6 and 3 separately to get $\dfrac{36}{9}$, then making an arithmetic error. Choice B (16) happens when students correctly find that $p-1 = 6$, but then square 6 before dividing by 3, getting $\dfrac{36}{3} = 12$—though that's not 16 either. Choice B likely comes from squaring 4 instead of 2. Choice C ($\dfrac{8}{3}$) results from forgetting to square the entire fraction and instead just doubling it. Remember: when evaluating expressions with substitution, always work step-by-step through the order of operations. Simplify completely before applying exponents to avoid fraction arithmetic errors.

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