Equations With One Variable - SAT Math

1. A shipping cost is modeled by $\frac{x}{4}+\frac{x}{6}=10$, where $x$ is the weight in pounds. What is the value of $x$?
   Answer: $24$ We need to solve $x/4 + x/6 = 10$ by finding a common denominator for the fractions. The LCD of 4 and 6 is 12, so we rewrite: $(3x/12) + (2x/12) = 10$, which gives us $5x/12 = 10$. Multiplying both sides by 12 yields $5x = 120$, so $x = 24$. A common error is to add denominators directly (getting $x/10 = 10$, thus $x = 100$) or to use 24 as the LCD instead of 12. When adding fractions, always find the least common denominator first.
2. Solve for $x$: $\frac{5x - 2}{3} = \frac{x + 10}{2}$.
   Answer: $\frac{34}{7}$ Cross-multiply to get $2(5x - 2) = 3(x + 10) \Rightarrow 10x - 4 = 3x + 30$, so $7x = 34$ and $x = 34/7$. The distractors reflect inverting the fraction, arithmetic slips, or sign errors.
3. Adult tickets cost $x$ dollars each, and child tickets cost $(x - 4)$ dollars each. If 3 adult tickets and 2 child tickets total $44, what is $x$?
   Answer: 10.4 Write $3x + 2(x - 4) = 44 \Rightarrow 5x - 8 = 44$, so $x = 52/5 = 10.4$. Using $x$ for both ticket types (8.8), treating children as $x + 4$ (7.2), or misdistributing the 2 (9.6) are common errors.
4. Solve the equation $8-(x-3)=2x+1$. What is the value of $x$? This problem tests subtracting a grouped expression: the minus sign in front of $(x-3)$ changes both signs inside the parentheses.
   Answer: $\frac{10}{3}$ We need to solve $8-(x-3)=2x+1$. First, distribute the negative sign: $8-x+3=2x+1$. Combine constants on the left: $11-x=2x+1$. Add $x$ to both sides: $11=3x+1$. Subtract 1: $10=3x$. Divide by 3: $x=\frac{10}{3}$. The key is remembering that the minus sign in front of $(x-3)$ changes both signs inside: $-(x-3)=-x+3$. This is a common source of errors.
5. Solve for $x$: $\frac{3}{4}x - 5 = 1$.
   Answer: 8 Add 5 to get $\frac{3}{4}x = 6$, then multiply by $\frac{4}{3}$ to find $x = 8$. The distractors come from adding or scaling incorrectly.
6. A store applies a discount and then adds a fixed shipping cost. The final price satisfies $\frac{3}{4}(x-8)+5=17$, where $x$ is the original price in dollars. What is the value of $x$?
   Answer: $24$ We need to solve $\frac{3}{4}(x-8) + 5 = 17$ for the original price $x$. First, subtract 5 from both sides: $\frac{3}{4}(x-8) = 12$. Next, multiply both sides by $\frac{4}{3}$ to eliminate the fraction: $x - 8 = 12 × \frac{4}{3} = 16$. Finally, add 8 to both sides: $x = 16 + 8 = 24$. The original price was $24. A common error is to distribute $\frac{3}{4}$ before isolating the parentheses, which creates unnecessary fractions. When a fraction multiplies an entire expression, isolate that expression first before clearing the fraction.
7. A science formula is written as $P=\dfrac{F}{A}$, where $P$ is pressure, $F$ is force, and $A$ is area. Solve this literal equation for $A$ in terms of $F$ and $P$. Choose the correct rearrangement.
   Answer: $A=\dfrac{F}{P}$ This problem asks us to solve the literal equation P = F/A for A. To isolate A, we multiply both sides by A: PA = F. Then we divide both sides by P: A = F/P. The key insight is that when a variable is in the denominator, we can multiply both sides by that variable to move it to the numerator. A common mistake is to incorrectly invert the fraction or confuse which variables go in the numerator versus denominator.
8. Solve for $x$: $4(2x - 3) = 5x + 9$.
   Answer: 7 Distribute to get $8x - 12 = 5x + 9$, so $3x = 21$ and $x = 7$. The other choices come from dividing by the wrong number or miscombining terms.
9. A store discounts an item by $15\%$, so the sale price is $0.85x$. If the sale price is $51$, what is the original price $x$? Solve the equation $0.85x=51$.
   Answer: $x=60$ This question asks for the original price x where the sale price after 15% discount is 0.85x = 51. Divide both sides by 0.85: x = 51 / 0.85. Calculating, 51 / 0.85 = 5100 / 85 = 60. A common error is using 0.15 instead of 0.85, like dividing 51 by 0.15 to get 340, which is irrelevant. Another mistake might be adding 15% incorrectly without setting up the equation. For test-taking, solve decimal equations by converting to fractions if needed, like 51 / (85/100) = 51 * (100/85) = 5100/85=60, and check: 0.85*60=51.
10. A linear function passes through $(2, 7)$ and $(8, -5)$. What is the value of $y$ when $x = 0$?
    Answer: 11 Slope $m = \frac{-5 - 7}{8 - 2} = -2$, so using $y = mx + b$ with $(2,7)$ gives $7 = -2(2) + b$, hence $b = 11$. Other choices come from sign errors in computing the slope or intercept.
11. A phone plan charges $x$ dollars per gigabyte plus a fixed $12$ dollars. Another plan charges $18$ dollars plus $0.5x$ dollars per gigabyte. If the two plans cost the same for a certain usage, the equation is $x + 12 = 18 + 0.5x$. What is the value of $x$?
    Answer: $12$ The equation $x + 12 = 18 + 0.5x$ represents when two phone plans cost the same. To solve, subtract $0.5x$ from both sides: $x - 0.5x + 12 = 18$, which gives us $0.5x + 12 = 18$. Subtract 12 from both sides: $0.5x = 6$. Divide by 0.5 (or multiply by 2): $x = 12$. The common error is incorrectly combining $x - 0.5x$ as $-0.5x$ instead of $0.5x$. Remember that $x = 1x$, so $1x - 0.5x = 0.5x$.
12. Solve for $x$: $5(2x - 1) = 3x + 19$.
    Answer: $\frac{24}{7}$ Distribute to get $10x - 5 = 3x + 19$, so $7x = 24$ and $x = \frac{24}{7}$. The distractors reflect inverting the fraction, sign errors, or failing to combine like terms correctly.
13. A rectangle has perimeter $P$ and length $L$ and width $W$, with $P=2L+2W$. Solve this equation for $W$ in terms of $P$ and $L$.
    Answer: $W=\frac{P-2L}{2}$ We need to solve $P = 2L + 2W$ for $W$ in terms of $P$ and $L$. First, subtract $2L$ from both sides: $P - 2L = 2W$. Then divide both sides by 2: $W = \frac{P - 2L}{2}$. A common error is to divide only one term by 2, getting $W = \frac{P}{2} - 2L$, or to incorrectly rearrange terms. When solving for a variable, perform the same operation on all terms to maintain equality.
14. To a 40-liter solution that is 25% acid, pure acid is added to obtain a solution that is 30% acid. How many liters of pure acid are added?
    Answer: $\frac{20}{7}$ Let $x$ be liters added; $\frac{10 + x}{40 + x} = 0.30$ leads to $10 + x = 12 + 0.30x$, so $0.70x = 2$ and $x = \frac{20}{7}$. Other options come from using the wrong percent or rounding.
15. A shipping company uses the formula $C=2.5w+4$ for cost $C$ (in dollars) based on package weight $w$ (in pounds). If a package costs $19$ dollars to ship, what is the value of $w$?
    Answer: $6$ Given $C = 2.5w + 4$ and $C = 19$, we need to find the weight $w$. Substituting 19 for C: $19 = 2.5w + 4$. Subtract 4 from both sides: $15 = 2.5w$. Divide both sides by 2.5: $w = 15 ÷ 2.5 = 6$. The package weighs 6 pounds. A common mistake is to divide 19 by 2.5 without first subtracting the base fee of $4. Remember that in linear cost models, there's often a fixed component that must be accounted for before finding the variable rate.
16. Solve for $x$: $\frac{3}{4}(x - 2) - \frac{1}{3}(x + 4) = 5$.
    Answer: $\frac{94}{5}$ Combine terms to get $\left(\frac{5}{12}\right)x - \frac{17}{6} = 5$, so $\left(\frac{5}{12}\right)x = \frac{47}{6}$ and $x = \frac{47}{6}\cdot\frac{12}{5} = \frac{94}{5}$. Distractors reflect miscombining constants or dividing by the reciprocal incorrectly.
17. Solve for $x$: $\dfrac{x-4}{3} + \dfrac{x+2}{6} = 5$. You will need a common denominator and careful combining of fractions before isolating the variable. What is the value of $x$?
    Answer: $12$ To solve $\frac{x-4}{3} + \frac{x+2}{6} = 5$, we need a common denominator of 6. Multiply the first fraction by $\frac{2}{2}$: $\frac{2(x-4)}{6} + \frac{x+2}{6} = 5$. This gives us $\frac{2x-8+x+2}{6} = 5$, which simplifies to $\frac{3x-6}{6} = 5$. Multiply both sides by 6: $3x - 6 = 30$. Add 6 to both sides: $3x = 36$. Divide by 3: $x = 12$. The common mistake is incorrectly finding the common denominator or forgetting to multiply the numerator when adjusting fractions. Always check your work by substituting back into the original equation.
18. A taxi charges $4 plus $2 per mile. If the total cost was $18, how many miles were driven?
    Answer: 7 Set up $2m + 4 = 18$, so $2m = 14$ and $m = 7$. The other choices result from subtracting incorrectly (6 or 8) or ignoring the base fee (9).
19. A scientist rearranges $P=\frac{F}{A}+2$ to solve for $F$, where $P$, $F$, and $A$ are variables. Solve for $F$ in terms of $P$ and $A$.
    Answer: $F=A(P-2)$ We need to solve $P=\frac{F}{A}+2$ for $F$. First, subtract 2 from both sides: $P-2=\frac{F}{A}$. Then multiply both sides by $A$: $A(P-2)=F$, or $F=A(P-2)$. The key is to isolate the fraction first by subtracting 2, then clear the denominator by multiplying. A common error is to multiply by $A$ before subtracting 2, which would give $AP=F+2A$.
20. Solve the literal equation $A=\frac{1}{2}bh$ for $h$ in terms of $A$ and $b$. Assume $b\neq 0$. Which expression is correct?
    Answer: $h=\frac{2A}{b}$ We need to solve $A=\frac{1}{2}bh$ for $h$ in terms of $A$ and $b$. To isolate $h$, we need to eliminate the coefficient $\frac{1}{2}b$. Multiply both sides by 2: $2A = bh$. Now divide both sides by $b$ (given that $b \neq 0$): $\frac{2A}{b} = h$, so $h = \frac{2A}{b}$. A common error is forgetting to multiply by 2 first, which would give the incorrect answer $h = \frac{A}{b}$. When solving literal equations, treat the variables you're not solving for as constants.

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