One- and Two-Step Equations - SHSAT Math

Solve for $y$ in $3y = -12$ .
Answer: -4 When you encounter a linear equation like $3y = -12$ , you're dealing with one of the most fundamental algebra skills: isolating a variable. The goal is to get $y$ by itself on one side of the equation. To solve $3y = -12$ , you need to undo the multiplication of $y$ by 3. Since multiplication and division are inverse operations, divide both sides by 3: $\frac{3y}{3} = \frac{-12}{3}$ The 3's cancel on the left side, leaving $y = -4$ . Let's examine why the other answers are incorrect: A) -4 is correct, as shown above. B) 4 represents a sign error. You might get this if you incorrectly think that dividing a negative number by a positive number gives a positive result. Remember: $\frac{-12}{3} = -4$ , not $+4$ . C) -9 suggests you subtracted 3 from -12 instead of dividing by 3. This shows confusion between the operations needed to solve $3y = -12$ versus $y + 3 = -12$ . D) 36 indicates you multiplied both sides by 3 instead of dividing: $3y \times 3 = -12 \times 3$ , giving $9y = -36$ . This moves you further from the solution rather than closer. Strategy tip: Always verify your answer by substituting back into the original equation. Here: $3(-4) = -12$ ✓. This quick check catches sign errors and operation mistakes, which are the most common traps in basic equation solving.
The sum of one-third of a number $n$ and seven is equal to the difference when 1 is subtracted from one-half of the same number. What is the value of $n$ ?
Answer: 48 Translate the words into a mathematical equation. 'One-third of a number $n$ and seven' is $\frac{1}{3}n + 7$ . 'The difference when 1 is subtracted from one-half of the same number' is $\frac{1}{2}n - 1$ . Set these two expressions equal: $\frac{1}{3}n + 7 = \frac{1}{2}n - 1$ . To solve this equation, first eliminate the fractions by multiplying the entire equation by the least common multiple of 3 and 2, which is 6. $6(\frac{1}{3}n) + 6(7) = 6(\frac{1}{2}n) - 6(1)$ , which simplifies to $2n + 42 = 3n - 6$ . This is a two-step equation. Subtract $2n$ from both sides: $42 = n - 6$ . Add 6 to both sides: $48 = n$ . The value of $n$ is 48. The distractors come from common errors, such as reversing the subtraction ( $1 - \frac{1}{2}n$ ) or making sign errors when clearing fractions.
A rental car company charges a flat fee of $45 plus $0.15 per mile driven. A second company charges a flat fee of $30 plus $0.20 per mile driven. For what number of miles driven is the cost from both companies exactly the same?
Answer: 300 miles Let $m$ be the number of miles driven. The cost for the first company, $C_1$ , can be modeled by the equation $C_1 = 45 + 0.15m$ . The cost for the second company, $C_2$ , can be modeled by $C_2 = 30 + 0.20m$ . To find when the cost is the same, we set $C_1 = C_2$ : $45 + 0.15m = 30 + 0.20m$ . This is a two-step equation. First, subtract (0.15m) from both sides: $45 = 30 + 0.05m$ . Next, subtract 30 from both sides: $15 = 0.05m$ . To solve for $m$ , divide by 0.05: $m = \frac{15}{0.05} = \frac{1500}{5} = 300$ . The cost will be the same at 300 miles. The distractors represent common calculation errors, such as misplacing a decimal or subtracting costs instead of miles.
The average (arithmetic mean) of three numbers, $a$ , $b$ , and $c$ , is $M$ . The average of $a$ and $b$ is $N$ . If $c = \frac{1}{2}M$ , what is $N$ in terms of $M$ ?
Answer: $M(5/4)$ From the given information, we can write two equations. First, $\frac{a+b+c}{3} = M \implies a+b+c = 3M$ . Second, $\frac{a+b}{2} = N \implies a+b = 2N$ . We can substitute the second equation into the first: $2N + c = 3M$ . We are also given that $c = \frac{1}{2}M$ . Substitute this into the equation: $2N + \frac{1}{2}M = 3M$ . This is a two-step equation to solve for $N$ . First, subtract $\frac{1}{2}M$ from both sides: $2N = 3M - \frac{1}{2}M = \frac{6}{2}M - \frac{1}{2}M = \frac{5}{2}M$ . Second, divide both sides by 2: $N = \frac{5}{4}M$ . The distractors represent common algebraic errors, such as incorrectly combining terms or isolating the wrong variable.
A cell phone plan costs $30 per month plus $8 per gigabyte (GB) for data used beyond a 2 GB allowance. A 15% tax is calculated on the entire monthly bill. If a customer's total bill after tax was $50.60, how many total gigabytes of data did they use that month?
Answer: 3.75 GB Let $d$ be the number of additional gigabytes used (beyond the 2 GB allowance). The cost before tax is the base fee plus the overage charge: $C = 30 + 8d$ . The total bill is this cost plus a 15% tax, which means multiplying by 1.15. So, $50.60 = 1.15(30 + 8d)$ . This is a two-step equation. First, divide by 1.15: $\frac{50.60}{1.15} = 44$ . So, $44 = 30 + 8d$ . Next, subtract 30: $14 = 8d$ . Solve for $d$ : $d = \frac{14}{8} = \frac{7}{4} = 1.75$ . This is the additional data used. The question asks for the total data, which is the 2 GB allowance plus the additional data: $2 + 1.75 = 3.75$ GB. Distractor A is the amount of additional data, not the total. Distractors B and D result from common errors in calculating with percentages or setting up the initial equation.
Two cars, A and B, start at the same point and travel in the same direction. Car A travels at a constant speed of 50 miles per hour. Car B starts 30 minutes later and travels at a constant speed of 60 miles per hour. How many hours after Car A starts will Car B catch up to Car A?
Answer: 3.0 hours Let $t$ be the number of hours Car A has been traveling. When Car B catches up, they will have traveled the same distance. The distance for Car A is $D_A = 50t$ . Car B starts 30 minutes (0.5 hours) later, so its travel time is $t - 0.5$ . The distance for Car B is $D_B = 60(t - 0.5)$ . Set the distances equal: $50t = 60(t - 0.5)$ . Distribute on the right side: $50t = 60t - 30$ . This is a two-step equation. Subtract $60t$ from both sides: $-10t = -30$ . Divide by -10: $t = 3$ . So, Car B will catch up to Car A 3 hours after Car A starts. Distractor C is the time that Car B has been traveling ( $3 - 0.5 = 2.5$ hours). Distractors A and B result from errors in setting up the time relationship, for example using $t+0.5$ or making a calculation mistake.
If $3(x - 4) + 2x = 18$ , what is the value of $2x + 1$ ?
Answer: 13 First solve for x: 3(x - 4) + 2x = 18 → 3x - 12 + 2x = 18 → 5x - 12 = 18 → 5x = 30 → x = 6. Then 2x + 1 = 2(6) + 1 = 13. Choice A results from finding x - 1 instead of 2x + 1. Choice B results from solving incorrectly to get x = 3. Choice D results from finding 2x + 3 instead of 2x + 1.
If $\frac{3}{5}(2x - 7) = 9$ , what is the value of $4x - 14$ ?
Answer: 30 We can solve this problem in two ways. Method 1: Solve for $x$ first. Multiply both sides of $\frac{3}{5}(2x - 7) = 9$ by $\frac{5}{3}$ : $2x - 7 = 9 \cdot \frac{5}{3} = 3 \cdot 5 = 15$ . This is a two-step equation. Add 7 to both sides: $2x = 22$ . Divide by 2: $x=11$ . Now substitute this into the expression $4x - 14$ : $4(11) - 14 = 44 - 14 = 30$ . Method 2: Notice the relationship between the expressions. The expression we want to find, $4x - 14$ , is exactly twice the expression inside the parentheses, $2x-7$ . From the first step of Method 1, we found that $2x - 7 = 15$ . Therefore, $4x-14 = 2(2x-7) = 2(15) = 30$ . Distractor A is the value of $x$ . Distractor B is the value of $2x-7$ . Distractor C is the value of $2x$ .
If $x$ and $y$ are positive integers such that $3x-7=14$ and $2xy=84$ , what is the value of $y$ ?
Answer: 6 This is a two-part problem. First, we must solve the one-step equation for $x$ . Given $3x-7=14$ , add 7 to both sides to get $3x = 21$ . Then divide by 3 to find $x=7$ . Now, substitute this value of $x$ into the second equation: $2(7)y = 84$ . This simplifies to $14y = 84$ . To find $y$ , divide both sides by 14: $y = \frac{84}{14} = 6$ . The value of $y$ is 6. Distractor B is the value of $x$ . Distractor C is the value of $xy$ . Distractor D is the coefficient of $y$ in the second step.
At a certain store, the price of a jacket is reduced by 20%. A sales tax of 5% is then applied to the discounted price. If the final price after the tax is $113.40, what was the original price of the jacket?
Answer: $135.00 Let $P$ be the original price of the jacket. A 20% reduction means the discounted price is $100\% - 20\% = 80\%$ of the original price, or $0.80P$ . A 5% sales tax is applied to this discounted price, which is equivalent to multiplying by 1.05. The final price is $1.05(0.80P)$ . We are given that this final price is $113.40. So, $1.05(0.80P) = 113.40$ . First, multiply the decimals on the left: $1.05 \times 0.80 = 0.84$ . The equation is now $0.84P = 113.40$ . This is a one-step equation. To find $P$ , divide both sides by 0.84: $P = \frac{113.40}{0.84} = 135$ . The original price was $135.00. Distractors often come from reversing the operations: dividing by 1.05 first and then by 0.80, or incorrectly adding/subtracting percentages.
A number is tripled, then 8 is subtracted from the result. If this final result equals the original number increased by 4, what was the original number?
Answer: 6 Let x = the original number. The equation is: 3x - 8 = x + 4. Solving: 3x - x = 4 + 8 → 2x = 12 → x = 6. Verify: Triple 6 gives 18, subtract 8 gives 10. Original number plus 4: 6 + 4 = 10 ✓. Choice A results from solving 2x = 4 instead of 2x = 12. Choice B results from confusing the final result with the original number. Choice D results from the value 4 + 8 = 12, but forgetting to divide by 2.
What is the value of $n$ that satisfies the equation $\frac{3n - 5(n-2)}{4} = \frac{1}{2} - n$ ?
Answer: -4 First, simplify the numerator on the left side of the equation: $3n - 5(n-2) = 3n - 5n + 10 = -2n + 10$ . The equation becomes $\frac{-2n+10}{4} = \frac{1}{2} - n$ . The fraction on the left can be simplified by dividing the numerator and denominator by 2: $\frac{-n+5}{2}$ . So, $\frac{-n+5}{2} = \frac{1}{2} - n$ . To eliminate the denominators, multiply the entire equation by 2: $-n+5 = 1 - 2n$ . Now, this is a two-step equation. Add $2n$ to both sides: $n+5 = 1$ . Subtract 5 from both sides: $n = -4$ . Distractor A results from a sign error in the initial distribution: $3n-5n-10$ instead of $3n-5n+10$ , leading to $n=6$ . Distractor C comes from incorrectly moving terms, e.g., $-n+2n = 1+5$ . Distractor D is a sign error in the final step.
The sum of three consecutive integers is 54. What is the largest of these three integers?
Answer: 19 Let the three consecutive integers be n, n+1, and n+2. Their sum is: n + (n+1) + (n+2) = 54 → 3n + 3 = 54 → 3n = 51 → n = 17. Therefore, the three integers are 17, 18, and 19, making the largest one 19. Choice A gives the smallest integer instead of the largest. Choice B gives the middle integer. Choice D results from setting up the equation as 3n = 54, giving n = 18, then incorrectly adding 2 to get the largest.
Let the operation $\$$ be defined for any number $n$ as $\$n = \frac{2n-5}{3}$ . If $\$(\$(p)) = p-5$ , what is the value of $p$ ?
Answer: 4 This problem involves a nested application of a defined operation. First, find an expression for $\$(\$(p))$ . $\$p = \frac{2p-5}{3}$ . Now, apply the operation again to this result: $\$(\$(p)) = \$(\frac{2p-5}{3}) = \frac{2(\frac{2p-5}{3})-5}{3}$ . To simplify the numerator: $\frac{2(2p-5)}{3} - \frac{15}{3} = \frac{4p-10-15}{3} = \frac{4p-25}{3}$ . So, $\$(\$(p)) = \frac{4p-25}{9}$ . We are given that this equals $p-5$ . So, $\frac{4p-25}{9} = p-5$ . Multiply by 9: $4p-25 = 9(p-5) = 9p-45$ . Add 45 to both sides: $4p+20 = 9p$ . Subtract $4p$ from both sides: $20 = 5p$ . Divide by 5: $p=4$ . Distractor D comes from setting $\$p = p-5$ instead of the nested operation. Distractor C results from an algebraic error in simplifying the nested function. Distractor B is close to the correct answer.
If $5x - 2(x + 4) = 3x - 16$ , what is the value of $x$ ?
Answer: No solution exists for this equation Expand the left side: 5x - 2(x + 4) = 5x - 2x - 8 = 3x - 8. The equation becomes 3x - 8 = 3x - 16. Subtracting 3x from both sides: -8 = -16, which is false. Therefore, no solution exists. Choice A would occur if both sides simplified to identical expressions. Choice C comes from incorrectly solving as if -8 = -16 + 2x, giving x = 4. Choice D comes from a sign error in the expansion or solving process.
At a school fundraiser, adult tickets cost $12 and student tickets cost $7. The total revenue from 150 tickets was $1400. To find the number of student tickets, $s$ , the equation $12(150-s) + 7s = 1400$ is used. Which of the following is the correct next step in solving this equation?
Answer: $1800 - 12s + 7s = 1400$ The question asks for the correct next step in solving the given equation $12(150-s) + 7s = 1400$ . The first step in simplifying this equation is to apply the distributive property to the term $12(150-s)$ . This means multiplying 12 by both 150 and $-s$ . $12 \times 150 = 1800$ and $12 \times (-s) = -12s$ . Substituting this back into the equation gives $1800 - 12s + 7s = 1400$ . This matches choice A. Choice B incorrectly combines $-s$ and $7s$ before distributing. Choice C incorrectly combines $-12s$ and $7s$ into $-5s$ without distributing the 12 to 150 first. Choice D incorrectly adds $-12s$ and $-7s$ instead of $7s$ .
The sum of the ages of three siblings, Ann, Bob, and Chloe, is 52. Ann is 5 years younger than Bob. Chloe's age is 3 years more than twice Ann's age. How old is Bob?
Answer: 16 Let $A$ , $B$ , and $C$ be the ages of Ann, Bob, and Chloe, respectively. We are given three relationships: 1) $A+B+C = 52$ . 2) $A = B - 5$ . 3) $C = 2A + 3$ . Our goal is to find $B$ . We can express all ages in terms of a single variable. Since A is in both (2) and (3), let's use A. From (2), we can write $B = A+5$ . Now substitute the expressions for $B$ and $C$ into the first equation: $A + (A+5) + (2A+3) = 52$ . Combine like terms: $4A + 8 = 52$ . This is a two-step equation. Subtract 8 from both sides: $4A = 44$ . Divide by 4: $A = 11$ . This is Ann's age. The question asks for Bob's age. Using $B = A+5$ , we find Bob's age is $B = 11+5 = 16$ . Distractor A is Ann's age. Distractor C is Chloe's age ( $2(11)+3=25$ ). Distractor D is a calculation error.
What value of $x$ satisfies the equation $x + 7 = 15$ ?
Answer: 8 When you encounter a linear equation like $x + 7 = 15$ , you're working with one of the most fundamental concepts in algebra: isolating the variable. Your goal is to get $x$ by itself on one side of the equation. To solve this, you need to "undo" what's being done to $x$ . Since 7 is being added to $x$ , you subtract 7 from both sides to maintain the equation's balance: $x + 7 = 15$ $x + 7 - 7 = 15 - 7$ $x = 8$ You can verify this by substituting back: $8 + 7 = 15$ ✓ Looking at the wrong answers reveals common mistakes: Choice B (22) comes from adding instead of subtracting: $15 + 7 = 22$ . This happens when students forget they need to do the opposite operation to isolate the variable. Choice C (-8) results from subtracting 7 from the left side but forgetting to subtract it from the right side, or from incorrectly thinking $x = 7 - 15$ . Choice D (108) seems to come from an entirely different calculation, possibly multiplying the numbers instead of using proper algebraic steps. Study tip: Always remember the golden rule of equations: whatever you do to one side, you must do to the other. When solving $x + a = b$ , subtract $a$ from both sides to get $x = b - a$ . Always check your answer by substituting it back into the original equation.
A rectangle's length is $5x-4$ and its width is $2x+1$ . An equilateral triangle's side is $6x-5$ . The perimeters of the two shapes are equal. What is the area of a square whose side length is equal to the value of $x$ ?
Answer: $\frac{81}{16}$ First, set up expressions for the perimeters. The perimeter of the rectangle is $2(L+W) = 2((5x-4) + (2x+1)) = 2(7x-3) = 14x-6$ . The perimeter of the equilateral triangle is $3s = 3(6x-5) = 18x-15$ . Set the perimeters equal: $14x-6 = 18x-15$ . This is a two-step equation. Add 15 to both sides: $14x+9 = 18x$ . Subtract $14x$ from both sides: $9 = 4x$ , so $x = \frac{9}{4}$ . The question asks for the area of a square with side length $x$ . The area is $x^2 = (\frac{9}{4})^2 = \frac{81}{16}$ . Distractor A is the value of $x$ . Distractor B is the value of the perimeter ( $14(\frac{9}{4})-6 = \frac{63}{2}-6 = \frac{51}{2}$ ). Distractor D is the side length of the triangle ( $6(\frac{9}{4})-5 = \frac{27}{2}-5 = \frac{17}{2}$ ).

Start this quiz