Equations With One Variable - PSAT Math

A store owner buys an item for $c$ dollars. To set a selling price, the owner marks up the cost by 50%. After two weeks, the item has not sold, so the owner offers a 20% discount off the current selling price. If the final price after the discount is $\$36, what was the original cost,$ c$$?
Answer: $\$30.00 When you encounter markup and discount problems, you're working with percentage changes applied in sequence. The key is to track how the price changes step by step, then work backwards from the final price. Let's trace through the price changes. The store owner starts with cost$ c $, then marks it up 50%. This gives a selling price of$ c + 0.50c = 1.50c $. Next comes a 20% discount off this selling price:$ 1.50c - 0.20(1.50c) = 1.50c(1 - 0.20) = 1.50c \times 0.80 = 1.20c $. Since the final price is \$36, we have$ 1.20c = 36 $, so$ c = 36 ÷ 1.20 = 30$$. The original cost was $30. Looking at the wrong answers: Choice A ($22.50) represents a common error where students might subtract both percentages from 100% incorrectly, thinking the final multiplier is 0.30 instead of 1.20. Choice C ($37.50) could result from confusing which percentage to apply when, perhaps treating the discount as applying to the original cost. Choice D ($45.00) might come from incorrectly adding the percentages or applying them in the wrong direction. The correct answer is B. Study tip: For sequential percentage problems, convert each percentage to a multiplier (50% markup means multiply by 1.50, 20% discount means multiply by 0.80), then combine them before solving. This prevents calculation errors and makes the algebra cleaner.
If $\frac{x+1}{2} + \frac{x+2}{3} = \frac{16}{3}$ , what is the value of $x$ ?
Answer: 5 When you encounter an equation with fractions, your goal is to clear the denominators and solve for the variable systematically. To solve $\frac{x+1}{2} + \frac{x+2}{3} = \frac{16}{3}$ , find a common denominator for the left side. The LCD of 2 and 3 is 6, so multiply each term appropriately: $\frac{3(x+1)}{6} + \frac{2(x+2)}{6} = \frac{16}{3}$ This simplifies to: $\frac{3x+3+2x+4}{6} = \frac{16}{3}$ $\frac{5x+7}{6} = \frac{16}{3}$ Cross-multiply to eliminate fractions: $3(5x+7) = 6 \cdot 16$ $15x + 21 = 96$ $15x = 75$ $x = 5$ You can verify: $\frac{5+1}{2} + \frac{5+2}{3} = \frac{6}{2} + \frac{7}{3} = 3 + \frac{7}{3} = \frac{16}{3}$ ✓ Choice A ( $-\frac{1}{3}$ ) likely results from sign errors during fraction manipulation. Choice B (1) might come from incorrectly combining like terms or making arithmetic mistakes when finding the common denominator. Choice D (25) could result from errors in cross-multiplication, such as multiplying $15x = 75$ incorrectly or confusing the final division step. Strategy tip: When solving equations with multiple fractions, always find a common denominator first, then cross-multiply to eliminate fractions entirely. This reduces the chance of arithmetic errors and makes the algebra more straightforward. Always substitute your answer back into the original equation to verify it works.
A taxi fare is modeled by $3.50+2.25x=21.50$ , where $x$ is the number of miles. What is the value of $x$ ?
Answer: $8$ The problem asks to solve the taxi fare equation 3.50 + 2.25x = 21.50 for x, where x represents miles traveled. Subtract 3.50 from both sides: 2.25x = 18. Divide both sides by 2.25: x = 18 ÷ 2.25 = 8. A common error would be incorrect decimal division or forgetting to subtract the base fare first. When solving real-world linear equations, identify the fixed cost and variable cost components before isolating the variable.
Solve for $x$ : $\dfrac{3}{5}x+\dfrac{1}{2}=\dfrac{1}{10}x+\dfrac{7}{2}$ . What is the value of $x$ ?
Answer: $6$ The problem asks to solve (3/5)x + 1/2 = (1/10)x + 7/2 for x. Subtract (1/10)x from both sides: (3/5 - 1/10)x + 1/2 = 7/2. Convert to common denominator: (6/10 - 1/10)x = 7/2 - 1/2, so (5/10)x = 6/2, which gives (1/2)x = 3. Finally, multiply by 2: x = 6. A key error would be incorrect fraction arithmetic when finding common denominators. Always convert fractions to common denominators before combining.
What is the solution to the equation $5x+12=3x-8$ ? The equation has variables on both sides, so isolate $x$ carefully.
Answer: $-10$ The problem asks to solve 5x + 12 = 3x - 8 for x, which has variables on both sides. Subtract 3x from both sides: 2x + 12 = -8. Subtract 12 from both sides: 2x = -20. Divide both sides by 2: x = -10. A key mistake would be incorrectly moving terms or making sign errors when combining like terms. When variables appear on both sides, collect all variable terms on one side and constants on the other.
What is the solution to the equation $3(x - 2(x + 1)) = 4(x - 5) + 2$ ?
Answer: $\frac{12}{7}$ This equation requires careful algebraic manipulation, particularly when dealing with nested parentheses. The key is to work systematically from the inside out, then combine like terms to isolate the variable. Start with the left side: $3(x - 2(x + 1))$ . First, distribute the $-2$ inside the inner parentheses: $x - 2(x + 1) = x - 2x - 2 = -x - 2$ . Now distribute the $3$ : $3(-x - 2) = -3x - 6$ . For the right side: $4(x - 5) + 2 = 4x - 20 + 2 = 4x - 18$ . The equation becomes: $-3x - 6 = 4x - 18$ Collect like terms by adding $3x$ to both sides: $-6 = 7x - 18$ Add $18$ to both sides: $12 = 7x$ Therefore: $x = \frac{12}{7}$ Choice A (3) likely comes from making errors in the distribution process and getting a simpler equation. Choice B ( $\frac{16}{7}$ ) probably results from a sign error when combining the constant terms—getting $16 = 7x$ instead of $12 = 7x$ . Choice C ( $\frac{16}{3}$ ) suggests confusion in both the coefficient of $x$ and the constant term, possibly mixing up the $7$ and $3$ from the distribution steps. When solving multi-step equations with nested parentheses, always distribute from the innermost parentheses outward, then carefully track positive and negative signs as you combine like terms. Double-check by substituting your answer back into the original equation.
Solve the equation $10-(3x+2)=2x-7$ . What is the value of $x$ ?
Answer: $3$ The problem asks to solve 10 - (3x + 2) = 2x - 7 for x. Distribute the negative: 10 - 3x - 2 = 2x - 7, which gives 8 - 3x = 2x - 7. Add 3x to both sides: 8 = 5x - 7, then add 7: 15 = 5x. Finally, divide by 5: x = 3. A key error would be incorrect distribution of the negative sign or arithmetic mistakes. When removing parentheses with a negative sign, change the sign of every term inside.
Solve the equation $4.5x-2=1.5x+10$ . What is the value of $x$ ?
Answer: $4$ The problem asks to solve 4.5x - 2 = 1.5x + 10 for x. Subtract 1.5x from both sides: 3x - 2 = 10, then add 2: 3x = 12. Finally, divide by 3: x = 4. A common error is decimal arithmetic mistakes or incorrect combining of like terms. When working with decimal coefficients, be extra careful with subtraction and division operations.
If $4x-\dfrac{1}{3}=\dfrac{5}{6}$ , what is the value of $x$ ? Combine the fractions on the right side before dividing.
Answer: $\dfrac{7}{24}$ The problem asks to solve 4x - 1/3 = 5/6 for x. Add 1/3 to both sides: 4x = 5/6 + 1/3. Convert to common denominator: 4x = 5/6 + 2/6 = 7/6. Divide by 4: x = 7/6 ÷ 4 = 7/24. A common error is incorrect fraction arithmetic when adding or dividing fractions. When combining fractions, always find a common denominator first, then divide carefully.
For what value of $a$ does the equation $5(x - 2) - x = 2(ax - 3)$ have no solution?
Answer: 2 When you encounter an equation with a parameter and are asked when it has "no solution," you're dealing with a situation where the equation becomes contradictory—like $3 = 5$ . Let's simplify both sides of $5(x - 2) - x = 2(ax - 3)$ and see what happens. On the left side: $5(x - 2) - x = 5x - 10 - x = 4x - 10$ . On the right side: $2(ax - 3) = 2ax - 6$ . So our equation becomes $4x - 10 = 2ax - 6$ . Rearranging to get all terms on one side: $4x - 2ax = -6 + 10$ , which gives us $(4 - 2a)x = 4$ . For this equation to have a unique solution, we'd divide both sides by $(4 - 2a)$ . But if $4 - 2a = 0$ , we can't divide, and we get $0 \cdot x = 4$ , or $0 = 4$ —which is impossible. This happens when $4 - 2a = 0$ , so $a = 2$ . Looking at the wrong answers: Choice A (-2) would give us $8x = 4$ , which has solution $x = \frac{1}{2}$ . Choice B (0) would give us $4x = 4$ , so $x = 1$ . Choice D (4) would give us $-4x = 4$ , so $x = -1$ . All of these produce valid solutions. Only choice C creates the contradiction that results in no solution. Strategy tip: When asked about "no solution" scenarios, look for cases where you end up with a false statement like $0 = \text{nonzero number}$ after simplifying.
If $0.6x-4.2=1.8x+1.8$ , what is the value of $x$ ? Make sure the sign is handled correctly when moving terms.
Answer: $-5$ The problem asks to solve $0.6x - 4.2 = 1.8x + 1.8$ for x. Subtract 0.6x from both sides: $-4.2 = 1.2x + 1.8$ . Subtract 1.8 from both sides: $-6 = 1.2x$ . Divide both sides by 1.2: $x = -6 \div 1.2 = -5$ . A key error would be incorrectly handling the signs when moving terms or making decimal arithmetic mistakes. When working with decimal coefficients, be extra careful with sign changes and division operations.
Solve the equation $\dfrac{2x-1}{3}=\dfrac{x+5}{2}$ . What is the value of $x$ ?
Answer: $17$ The problem asks to solve (2x - 1)/3 = (x + 5)/2 for x. Cross-multiply to get 2(2x - 1) = 3(x + 5), which gives 4x - 2 = 3x + 15. Subtract 3x from both sides: x - 2 = 15, then add 2: x = 17. A common error is making mistakes during cross-multiplication or not properly distributing. When solving rational equations, cross-multiplication is often the most efficient method to eliminate fractions.
A phone plan's cost in dollars is modeled by $12+0.75x=0.5x+18$ , where $x$ is the number of gigabytes used. What is the value of $x$ that makes the two sides equal?
Answer: $24$ We need to solve $12 + 0.75x = 0.5x + 18$ for the number of gigabytes $x$ . Subtracting $0.5x$ from both sides: $12 + 0.75x - 0.5x = 18$ , which simplifies to $12 + 0.25x = 18$ . Subtracting 12 from both sides: $0.25x = 6$ . Dividing by 0.25: $x = 6 ÷ 0.25 = 24$ . A common error is making arithmetic mistakes with decimals or incorrectly combining the $x$ terms. When working with decimal coefficients, consider converting to fractions to avoid calculation errors.
Solve for $x$ : $7x-3=2x+\dfrac{17}{2}$ . What is the value of $x$ ?
Answer: $\dfrac{23}{10}$ The problem asks to solve 7x - 3 = 2x + 17/2 for x. Subtract 2x from both sides: 5x - 3 = 17/2, then add 3: 5x = 17/2 + 6/2 = 23/2. Finally, divide by 5: x = 23/10. A common error is incorrect fraction arithmetic when combining terms. When adding fractions to whole numbers, convert the whole number to a fraction with the same denominator first.
A water tank starts with 120 liters and drains at $x$ liters per minute. After 9 minutes, 66 liters remain, modeled by $120-9x=66$ . What is $x$ ?
Answer: $6$ The problem describes a water tank draining equation 120 - 9x = 66, where 120 is initial volume, 9x is amount drained in 9 minutes at x liters per minute, and 66 is final volume. Subtract 120 from 66: -9x = 66 - 120 = -54. Divide by -9: x = 6. A common mistake would be incorrect subtraction or sign handling. When solving real-world problems, ensure the equation makes physical sense before solving.
If $\frac{2y+3}{5} - \frac{y-2}{2} = \frac{19 - (y-1)}{10}$ , what can be concluded about the value of $y$ ?
Answer: The equation has no solution. When you encounter an equation with fractions, your goal is to solve for the variable by clearing denominators and simplifying. However, sometimes the algebra reveals that no solution exists. To solve this equation, first find a common denominator. The LCD of 5, 2, and 10 is 10. Multiply every term by 10: $10 \cdot \frac{2y+3}{5} - 10 \cdot \frac{y-2}{2} = 10 \cdot \frac{19-(y-1)}{10}$ This gives you: $2(2y+3) - 5(y-2) = 19-(y-1)$ Expand the left side: $4y + 6 - 5y + 10 = 19 - y + 1$ Simplify: $-y + 16 = 20 - y$ Add $y$ to both sides: $16 = 20$ Since 16 never equals 20, this equation has no solution. Answer choice A ( $y = 1$ ) and B ( $y = 5$ ) are wrong because you can substitute these values back into the original equation and verify they don't work. These represent common computational errors students might make when solving. Answer choice D (true for all real numbers) would only occur if you simplified to an identity like $0 = 0$ or $x = x$ , which didn't happen here. Strategy tip: When solving linear equations, if you eliminate the variable and get a false statement (like $16 = 20$ ), the equation has no solution. If you get a true statement (like $5 = 5$ ), it's true for all real numbers. Always double-check by substituting potential answers back into the original equation.
A contractor charges $50 for a visit plus $x$ dollars per hour. If a 3-hour job costs $170, then $50+3x=170$ . What is $x$ ?
Answer: $40$ The problem describes a contractor's billing equation $50 + 3x = 170$ , where 50 is the visit fee and $3x$ represents 3 hours at x dollars per hour. Subtract 50 from both sides: $3x = 120$ , then divide by 3: $x = 40$ . A common mistake would be dividing by the wrong coefficient or making subtraction errors. When solving real-world linear equations, clearly identify the fixed and variable components before isolating the variable.
A streaming service bills $9 plus $x$ dollars per movie. If the bill for 5 movies is $24, then $9+5x=24$ . What is $x$ ?
Answer: $3$ The problem describes a streaming service billing equation 9 + 5x = 24, where 9 is the base fee and 5x represents 5 movies at x dollars each. Subtract 9 from both sides: 5x = 15, then divide by 5: x = 3. A common mistake would be dividing by the wrong coefficient or making arithmetic errors in subtraction. When solving real-world problems, identify the fixed and variable costs before isolating the variable.
A phone plan charges $18 plus $x$ dollars per gigabyte. If 7 gigabytes cost $46, the situation is modeled by $18+7x=46$ . What is the value of $x$ ?
Answer: $4$ The problem asks to find x from the phone plan equation $18 + 7x = 46$ , where 18 is the base charge and $7x$ represents 7 gigabytes at x dollars each. Subtract 18 from both sides: $7x = 46 - 18 = 28$ . Divide both sides by 7: $x = 28 \div 7 = 4$ . A common mistake would be dividing the wrong terms or making arithmetic errors in subtraction. Always isolate the variable term first, then divide by its coefficient.
A company's total cost $C$ , in dollars, to produce $x$ units of a product is given by the function $C(x) = 15x + 500$ . The company sells each unit for $\$25. What is the number of units the company must sell to achieve a profit of exactly$ $4,500?
Answer: 500 This problem tests your ability to set up and solve profit equations. When you see questions about company costs, revenues, and profits, remember that profit equals revenue minus costs. First, let's establish the key relationships. The cost function is $C(x) = 15x + 500$ , where $x$ is the number of units produced. Since each unit sells for $25, the revenue function is $R(x) = 25x$ . Profit is calculated as $P(x) = R(x) - C(x) = 25x - (15x + 500) = 10x - 500$ . To find when profit equals $4,500, set up the equation: $10x - 500 = 4500$ . Adding 500 to both sides gives $10x = 5000$ , so $x = 500$ . This confirms answer choice D is correct. Let's examine why the other answers are wrong. Choice A (300 units) would yield a profit of $10(300) - 500 = 2500$ , which is $2,000 short of the target. Choice B (400 units) gives $10(400) - 500 = 3500$ , still $1,000 below the goal. Choice C (450 units) results in $10(450) - 500 = 4000$ , which is $500 less than needed. Each incorrect answer represents a common calculation error: either mistakes in setting up the profit equation or arithmetic errors when solving. Study tip: Always write out the profit equation explicitly as Revenue - Costs before substituting numbers. This prevents confusion about which values represent income versus expenses, and double-check your arithmetic by substituting your answer back into the original profit equation.

Start this quiz