One- and Two-Step Equations - ISEE Upper Level Quantitative Reasoning

1. A rectangle has length $l$ and width $w$. If both the length and width are increased by 3 units, the new area is 39 square units more than the original area. Which equation represents this relationship?
   Answer: $(l + 3)(w + 3) = lw + 39$ When you encounter problems about changing dimensions and comparing areas, focus on setting up equations that accurately represent the relationship between original and new measurements. The original rectangle has area $lw$. After increasing both length and width by 3 units, the new dimensions become $(l + 3)$ and $(w + 3)$, giving a new area of $(l + 3)(w + 3)$. Since this new area is 39 square units more than the original area, you can write: new area = original area + 39, which translates to $(l + 3)(w + 3) = lw + 39$. Looking at the wrong answers: Choice B incorrectly adds the new dimensions $(l + 3) + (w + 3)$ instead of multiplying them. This gives you a perimeter-related expression, not area. Choice C simplifies the left side of choice B to $l + w + 6$, but this still represents a linear measurement, not area. Choice D rearranges the correct relationship incorrectly—it states $lw + 6 = (l + 3)(w + 3) - 39$, which would mean the original area plus 6 equals the new area minus 39. This doesn't match the problem statement. To verify choice A works, expand $(l + 3)(w + 3) = lw + 3l + 3w + 9$. The increase in area is $3l + 3w + 9 = 3(l + w + 3)$, which equals 39 according to the problem. Study tip: For area problems involving dimension changes, always write "new area = original area ± change" first, then substitute the appropriate formulas for each area.
2. If $5(x - 2) = 3(x + 4)$, what is the value of $2x + 1$?
   Answer: 23 This question tests your ability to solve linear equations and then evaluate expressions using the solution. When you see an equation with variables on both sides, your goal is to isolate the variable by collecting like terms. Start by expanding both sides: $5(x - 2) = 3(x + 4)$ becomes $5x - 10 = 3x + 12$. Now collect all $x$ terms on one side and constants on the other. Subtract $3x$ from both sides: $2x - 10 = 12$. Add $10$ to both sides: $2x = 22$, so $x = 11$. The question asks for $2x + 1$, not just $x$. Substitute $x = 11$: $2(11) + 1 = 22 + 1 = 23$. Looking at the wrong answers: Choice (A) gives 11, which is the value of $x$ itself—this catches students who stop solving too early and don't evaluate the requested expression. Choice (B) gives 21, which equals $2x - 1$ when $x = 11$; this traps students who make a sign error in the final calculation. Choice (D) gives 45, which doesn't correspond to any reasonable mistake in this problem. The key strategy here is to always read carefully what the question is asking for. Many algebra problems on standardized tests will ask you to find an expression involving the variable rather than the variable itself. After solving for the variable, take that extra step to substitute into the requested expression. This two-step process prevents careless errors and ensures you're answering the actual question.
3. Sarah has some money in her savings account. After depositing $15, she has twice as much as she would have if she had withdrawn $8 instead. How much money did Sarah have originally?
   Answer: $31 This is an algebraic word problem that requires you to set up an equation based on two different scenarios involving the same original amount. Let's call Sarah's original amount $x$. The problem gives us two scenarios: one where she deposits $15 (giving her $x + 15$), and another where she withdraws $8 (giving her $x - 8$). The key relationship is that after depositing $15, she has twice as much as she would have after withdrawing $8. Setting up the equation: $x + 15 = 2(x - 8)$ Solving: $x + 15 = 2x - 16$ $15 + 16 = 2x - x$ $31 = x$ So Sarah originally had $31, which is choice (B). Let's verify the wrong answers. Choice (A) $23 would give us: after depositing $15, she'd have $38, but twice what she'd have after withdrawing $8 would be $2(23-8) = 2(15) = 30$. Since $38 ≠ 30$, this is incorrect. Choice (C) $38 would give us: after depositing, she'd have $53, but twice the withdrawal scenario gives $2(38-8) = 2(30) = 60$. Since $53 ≠ 60$, this is wrong. Choice (D) $46 would give us: after depositing, she'd have $61, but twice the withdrawal scenario gives $2(46-8) = 2(38) = 76$. Since $61 ≠ 76$, this is incorrect. When solving word problems like this, always define your variable clearly, translate the relationship into an equation, and verify your answer by checking it against the original conditions.
4. Maria solved the equation $4(y + 3) = 2y - 6$ and got $y = -9$. When she checks her answer by substituting back into the original equation, what value does she get for the left side?
   Answer: -24 When checking algebraic solutions, you substitute your answer back into the original equation to verify it's correct. This process helps catch calculation errors and confirms your solution is valid. To find what Maria gets for the left side, substitute $y = -9$ into the left side of the original equation $4(y + 3) = 2y - 6$. The left side is $4(y + 3)$, so: $4(y + 3) = 4(-9 + 3) = 4(-6) = -24$ This confirms that when Maria substitutes $y = -9$ into the left side, she gets $-24$. Let's examine why the other answers are incorrect. Choice B (-6) represents the value inside the parentheses after substitution ($-9 + 3 = -6$), but fails to multiply by 4. Choice C (0) might come from incorrectly thinking the left and right sides should equal zero, or from calculation errors. Choice D (6) could result from a sign error, perhaps calculating $4(-9 + 3)$ as $4(6)$ instead of $4(-6)$. Note that if you also substitute $y = -9$ into the right side $2y - 6$, you get $2(-9) - 6 = -18 - 6 = -24$. Since both sides equal $-24$, Maria's solution $y = -9$ is indeed correct. Remember: when checking algebraic solutions, substitute carefully and follow order of operations. Work inside parentheses first, then multiply. This systematic approach prevents the sign errors and calculation mistakes that create wrong answer choices.
5. Jamie has 3 more than twice the number of stickers that Alex has. If Jamie has 17 stickers, how many stickers does Alex have?
   Answer: 7 stickers When you encounter word problems involving relationships between quantities, your first step is to translate the words into mathematical expressions. This question describes Jamie's stickers in terms of Alex's stickers, so you need to work backwards from Jamie's known quantity. Let's call Alex's number of stickers $x$. The phrase "3 more than twice the number" translates to $2x + 3$. Since Jamie has 17 stickers, you can write the equation: $2x + 3 = 17$ To solve for $x$, subtract 3 from both sides: $2x = 14$. Then divide by 2: $x = 7$. Therefore, Alex has 7 stickers. Let's check why the other answers don't work. Choice B (10 stickers) would give Jamie $2(10) + 3 = 23$ stickers, not 17. Choice C (14 stickers) would result in $2(14) + 3 = 31$ stickers for Jamie. Choice D (37 stickers) is what you'd get if you mistakenly calculated $2(17) + 3$, thinking you needed to find how many stickers Jamie would have if Alex had 17 — this reverses the relationship described in the problem. Choice A (7 stickers) is correct because $2(7) + 3 = 17$, which matches Jamie's actual count. Study tip: In "more than" problems, always identify which person or quantity is being described in terms of the other. Set up your variable for the unknown quantity, write the equation based on the given relationship, then solve systematically. Double-check by substituting your answer back into the original relationship.
6. A number decreased by 8 is equal to 3 times the sum of the number and 4. What is the number?
   Answer: -10 Translate each phrase into algebra. Let the number be $x$. "A number decreased by 8" is $x - 8$. "3 times the sum of the number and 4" is $3(x + 4)$. Set them equal and solve: $x - 8 = 3x + 12$ $-20 = 2x$ $x = -10$ Check: $-10 - 8 = -18$ and $3(-10 + 4) = 3(-6) = -18$. The value $-6$ comes from writing $3x + 4$, adding 4 after multiplying instead of multiplying the sum. The value $-2$ comes from $x - 8 = 3x - 4$, a sign error on the 4. The value $-8$ treats the "decreased by 8" as the answer without solving. Tip: "times the sum" means the parentheses come first; always substitute your answer back into both sides.
7. The equation $\frac{x + a}{3} = \frac{x - a}{2}$ is satisfied when $x = 10$. What is the value of $a$?
   Answer: $a = 2$ When you encounter an equation with two variables where one variable's value is given, you're looking at a substitution problem. The key is to substitute the known value and solve for the unknown. Since we know that $x = 10$ satisfies the equation $\frac{x + a}{3} = \frac{x - a}{2}$, let's substitute $x = 10$: $\frac{10 + a}{3} = \frac{10 - a}{2}$ To solve this, cross-multiply to eliminate the fractions: $2(10 + a) = 3(10 - a)$ $20 + 2a = 30 - 3a$ Now collect like terms by adding $3a$ to both sides and subtracting $20$ from both sides: $2a + 3a = 30 - 20$ $5a = 10$ $a = 2$ This confirms that choice B is correct. Let's check why the other answers don't work. Choice A ($a = 1$) would give us $\frac{11}{3} = \frac{9}{2}$, which simplifies to $\frac{22}{6} = \frac{27}{6}$ — not equal. Choice C ($a = 4$) yields $\frac{14}{3} = \frac{6}{2} = 3$, but $\frac{14}{3} ≠ 3$. Choice D ($a = 5$) gives us $\frac{15}{3} = 5$ and $\frac{5}{2} = 2.5$, so $5 ≠ 2.5$. Remember: when substituting values into equations, always cross-multiply to avoid fraction arithmetic errors, and verify your answer by plugging it back into the original equation.
8. Kevin's age is 4 years less than 3 times his sister's age. If Kevin is 14 years old, how old is his sister?
   Answer: 6 years old When you encounter word problems involving relationships between ages, your goal is to translate the verbal description into a mathematical equation, then solve for the unknown. Let's break down the given information: Kevin's age equals 3 times his sister's age minus 4 years. Since Kevin is 14, we can write this as an equation. Let $s$ represent his sister's age: $14 = 3s - 4$ To solve, add 4 to both sides: $18 = 3s$ Divide by 3: $s = 6$ So Kevin's sister is 6 years old. We can verify: 3 times 6 equals 18, and 18 minus 4 equals 14 ✓ Looking at the wrong answers: Choice B (8 years old) would make Kevin $3(8) - 4 = 20$ years old, not 14. Choice C (10 years old) would make Kevin $3(10) - 4 = 26$ years old. Choice D (18 years old) would make Kevin $3(18) - 4 = 50$ years old. These answers likely come from common algebraic mistakes, such as forgetting to subtract 4 or mixing up the operations in the original relationship. The correct answer is A. Strategy tip: Always define your variable clearly (what does the letter represent?), write the equation exactly as described in words, then solve step by step. Finally, plug your answer back into the original relationship to check that it produces the given information—this catches most algebraic errors.
9. For what value of $k$ does the equation $5x + k = 3x - 8$ have the solution $x = -2$?
   Answer: $k = -4$ If $x = -2$ is the solution, substituting it must make both sides equal. Left side: $5(-2) + k = -10 + k$. Right side: $3(-2) - 8 = -6 - 8 = -14$. Set them equal: $-10 + k = -14$, so $k = -4$. Check: $5(-2) + (-4) = -14$ and $3(-2) - 8 = -14$. The value $k = -14$ is the value of the right side, not of $k$. The value $k = 14$ drops the negative sign on that right side. The value $k = -2$ reuses the value of $x$ as the value of $k$; substituting it gives $-12$ on the left, not $-14$.
10. If $2(x + 3) = 5x - (x - 6)$, what is the value of $x$?
    Answer: $x = 0$ When you encounter an equation with variables on both sides, your goal is to isolate the variable by combining like terms and using inverse operations systematically. Start by expanding both sides of the equation $2(x + 3) = 5x - (x - 6)$. On the left side, distribute the 2: $2x + 6$. On the right side, distribute the negative sign carefully: $5x - x + 6$, which simplifies to $4x + 6$. Now you have $2x + 6 = 4x + 6$. Next, subtract 6 from both sides to eliminate the constant terms: $2x = 4x$. Then subtract $2x$ from both sides: $0 = 2x$. Dividing both sides by 2 gives you $x = 0$. Let's examine why the other answers are incorrect. Choice B ($x = 1$) would give you $2(4) = 8$ on the left and $5 - (-5) = 10$ on the right—these don't match. Choice C ($x = 3$) produces $2(6) = 12$ on the left and $15 - (-3) = 18$ on the right—again, unequal. Choice D ($x = 6$) yields $2(9) = 18$ on the left and $30 - 0 = 30$ on the right—also incorrect. The correct answer is A: $x = 0$. Strategy tip: Always check your answer by substituting it back into the original equation. Also, when distributing negative signs, be extra careful with the signs of each term—this is where many students make errors that lead to wrong answer choices.
11. If $3x - 7 = 2x + 5$, what is the value of $x - 4$?
    Answer: 8 When you encounter an equation like this, you're dealing with a two-step process: first solve for the variable, then use that value to answer what's actually being asked. To solve $3x - 7 = 2x + 5$, you need to isolate $x$ by getting all terms with $x$ on one side and all constants on the other. Subtract $2x$ from both sides: $3x - 2x - 7 = 2x - 2x + 5$, which gives you $x - 7 = 5$. Then add 7 to both sides: $x = 12$. Now here's the crucial part—the question asks for $x - 4$, not just $x$. Since $x = 12$, then $x - 4 = 12 - 4 = 8$. The correct answer is A. Looking at the wrong choices: B) 12 is the value of $x$ itself, which is a classic trap for students who solve correctly but forget to complete the final step. C) 16 would result from adding instead of subtracting: $x + 4 = 12 + 4 = 16$. D) -8 might come from sign errors during the solving process, such as incorrectly handling the subtraction when isolating $x$. The key strategy here is to always read the question twice—once before solving and once after. Many ISEE questions will ask for an expression involving your variable rather than the variable itself. After finding $x$, always double-check what the question is actually asking for before selecting your answer.
12. The equation $ax + 6 = 4x - 2$ has no solution when $a$ equals which value?
    Answer: $a = 4$ When you encounter an equation with a parameter that could result in "no solution," you're dealing with a scenario where the algebraic manipulation leads to a contradiction. Let's solve $ax + 6 = 4x - 2$ by collecting like terms. First, subtract $4x$ from both sides: $ax - 4x + 6 = -2$. Factor out $x$: $(a - 4)x + 6 = -2$. Subtract 6 from both sides: $(a - 4)x = -8$. Now here's the key insight: if $a - 4 = 0$ (meaning $a = 4$), our equation becomes $0 \cdot x = -8$, or $0 = -8$. This is impossible—no value of $x$ can make zero equal negative eight. Therefore, when $a = 4$, the equation has no solution. Let's check why the other options are wrong. For choice A, when $a = -4$: $(-4 - 4)x = -8$ becomes $-8x = -8$, so $x = 1$. This gives us a valid solution. For choice B, when $a = 0$: $-4x = -8$, so $x = 2$. Again, a valid solution exists. For choice C, when $a = 2$: $-2x = -8$, so $x = 4$. This also works. Only choice D creates the impossible situation where we need $0 = -8$. Strategy tip: When an equation has "no solution," look for cases where you end up with a coefficient of zero on the variable side but a non-zero constant on the other side. This creates a mathematical impossibility.
13. If $4x - 7 = 13 + 2x$, what is the value of $3x - 5$?
    Answer: 25 When you encounter an equation like this, you're being tested on solving linear equations and then using that solution in a different expression. The key insight is that you don't need to find $x$ explicitly—you can often manipulate the given equation to match what you're looking for. Let's solve $4x - 7 = 13 + 2x$ step by step. First, subtract $2x$ from both sides: $2x - 7 = 13$. Then add 7 to both sides: $2x = 20$. Therefore, $x = 10$. Now substitute $x = 10$ into $3x - 5$: $3(10) - 5 = 30 - 5 = 25$. This confirms answer choice B is correct. Let's examine why the other choices are wrong. Choice A (10) is simply the value of $x$ itself—this catches students who stop after solving for $x$ instead of evaluating the target expression. Choice C (30) represents $3x$ without subtracting the 5, which happens when you forget the final step. Choice D (35) might result from calculation errors, such as adding 5 instead of subtracting it ($30 + 5 = 35$). Strategy tip: On multi-step problems like this, always double-check that you're answering what the question actually asks for. Many students correctly solve for $x$ but then select that value instead of substituting it into the required expression. Make it a habit to reread the question after solving to ensure you've found what they want.
14. If $2(3x - 4) + 5 = 3x + 7$, what is the value of $x$?
    Answer: $x = \frac{10}{3}$ This is a linear equation that requires you to systematically apply the distributive property and combine like terms to isolate the variable. Start by distributing the 2 on the left side: $2(3x - 4) = 6x - 8$. The equation becomes $6x - 8 + 5 = 3x + 7$. Simplify the left side by combining the constants: $6x - 3 = 3x + 7$. Now isolate $x$ by moving all terms with $x$ to one side and constants to the other. Subtract $3x$ from both sides: $6x - 3x - 3 = 7$, which gives you $3x - 3 = 7$. Add 3 to both sides: $3x = 10$. Finally, divide by 3: $x = \frac{10}{3}$. Let's examine why the other answers are wrong. Choice A ($x = \frac{2}{3}$) likely results from incorrectly distributing or making arithmetic errors when combining fractions. Choice B ($x = 2$) probably comes from mishandling the distributive property or incorrectly moving terms across the equals sign. Choice D ($x = 4$) might occur if you forget to divide by the coefficient of $x$ at the final step, leaving you with $3x = 12$ instead of the correct $3x = 10$. You can verify: substituting $x = \frac{10}{3}$ into the original equation gives $2(3 \cdot \frac{10}{3} - 4) + 5 = 2(6) + 5 = 17$, and $3 \cdot \frac{10}{3} + 7 = 10 + 7 = 17$ ✓. Strategy tip: Always work systematically through distribution, combining like terms, and isolation steps. Double-check by substituting your answer back into the original equation.
15. If $\frac{x}{4} + 3 = \frac{x}{6} + 5$, what is the value of $x$?
    Answer: $x = 24$ This equation involves fractions with variables, so you'll need to isolate $x$ by eliminating the fractions and combining like terms. Start by moving all terms with $x$ to one side and constants to the other. Subtract $\frac{x}{6}$ from both sides and subtract 3 from both sides: $\frac{x}{4} - \frac{x}{6} = 5 - 3 = 2$ To subtract the fractions, find a common denominator. The LCD of 4 and 6 is 12: $\frac{3x}{12} - \frac{2x}{12} = 2$ $\frac{x}{12} = 2$ Multiply both sides by 12: $x = 24$ You can verify: $\frac{24}{4} + 3 = 6 + 3 = 9$ and $\frac{24}{6} + 5 = 4 + 5 = 9$ ✓ Choice A ($x = 12$) comes from incorrectly using 12 as the final answer instead of recognizing that $\frac{x}{12} = 2$ means $x = 24$. Choice B ($x = 18$) results from calculation errors when finding the common denominator or combining fractions. Choice D ($x = 30$) might come from using an incorrect LCD or making arithmetic mistakes during the solving process. Strategy tip: When solving equations with fractions, always find a common denominator before combining terms, and remember to multiply by the denominator in your final step to isolate the variable. Double-check your answer by substituting back into the original equation.
16. For which value of $c$ does the equation $3x + c = cx + 9$ have exactly one solution?
    Answer: Any value except $c = 3$ When you encounter a linear equation with a parameter like this, you need to determine when the equation behaves normally (one solution) versus when it becomes degenerate (no solutions or infinitely many solutions). Let's solve $3x + c = cx + 9$ by collecting like terms. Subtract $cx$ from both sides: $3x - cx + c = 9$. Factor out $x$ on the left: $(3-c)x + c = 9$. Now subtract $c$ from both sides: $(3-c)x = 9-c$. For a linear equation in the form $ax = b$, we get exactly one solution when $a \neq 0$ (specifically $x = \frac{b}{a}$). Here, that means we need $3-c \neq 0$, which gives us $c \neq 3$. When $c \neq 3$, we get the unique solution $x = \frac{9-c}{3-c}$. But what happens when $c = 3$? Substituting back: $(3-3)x = 9-3$, which becomes $0x = 6$. This is impossible—no value of $x$ makes this true, so there are no solutions. Choice A is wrong because $c = 0$ works fine—it gives us $3x = 9$, so $x = 3$. Choice C is backwards—when $c = 3$, we get no solutions, not one solution. Choice D is incorrect because $c = 9$ gives us $-6x = 0$, so $x = 0$, which is exactly one solution. Remember: for parametric linear equations, the coefficient of the variable cannot be zero if you want exactly one solution. Always check what happens when that coefficient equals zero.
17. A taxi charges a flat fee of $3 plus $0.50 per mile. If a trip costs $8.50, how many miles was the trip?
    Answer: 11 miles This is a classic linear equation problem where you have a fixed cost plus a variable cost. When you see taxi fares, phone bills, or rental fees, you're usually dealing with the format: Total Cost = Fixed Fee + (Rate × Quantity). Let's set up the equation. The taxi charges $3 flat fee plus $0.50 per mile, and the total trip costs $8.50. If we call the number of miles $m$, then: $8.50 = 3.00 + 0.50m$ To solve, subtract the flat fee from both sides: $8.50 - 3.00 = 0.50m$ $5.50 = 0.50m$ Divide both sides by 0.50: $m = \frac{5.50}{0.50} = 11$ So the trip was 11 miles, making A correct. Looking at the wrong answers: B (11.5 miles) would cost $3.00 + (11.5 × $0.50) = $8.75, which is too much. This might trick you if you made a small arithmetic error. C (17 miles) would cost $3.00 + (17 × $0.50) = $11.50 — this could result from forgetting to subtract the flat fee before dividing. D (23 miles) would cost $3.00 + (23 × $0.50) = $14.50, which might come from dividing the total cost by the rate without accounting for the flat fee at all. Remember: always isolate the variable portion first by subtracting any fixed costs, then solve for the variable. This two-step approach prevents the most common errors on these linear cost problems.

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