Solving Linear Equations - Accuplacer Aaf

1. A solution is 20% acid. Another solution is 50% acid. How many liters of the 50% solution must be added to 15 liters of the 20% solution to produce a new solution that is 40% acid?
   Answer: 30 Let $x$ be the number of liters of the 50% solution to be added. The total amount of acid in the final mixture is the sum of the amounts of acid from the two initial solutions. The amount of acid is the concentration times the volume. So, the equation is $0.20(15) + 0.50(x) = 0.40(15 + x)$. Calculate the known products: $3 + 0.50x = 0.40(15 + x)$. Distribute on the right side: $3 + 0.50x = 6 + 0.40x$. Subtract $0.40x$ from both sides: $3 + 0.10x = 6$. Subtract 3 from both sides: $0.10x = 3$. Finally, divide by 0.10: $x = \frac{3}{0.10} = 30$. So, 30 liters must be added.
2. The sum of two numbers is 36. One-half of the larger number is 3 more than the smaller number. What is the value of the larger number?
   Answer: 26 Let $L$ be the larger number and $S$ be the smaller number. The given information translates to two equations: 1) $L + S = 36$ and 2) $\frac{1}{2}L = S + 3$. From equation 1, we can express $S$ as $S = 36 - L$. Substitute this expression for $S$ into equation 2: $\frac{1}{2}L = (36 - L) + 3$. Simplify the right side: $\frac{1}{2}L = 39 - L$. To solve for $L$, add $L$ to both sides: $\frac{1}{2}L + L = 39$, which gives $\frac{3}{2}L = 39$. Finally, multiply both sides by $\frac{2}{3}$ to find $L$: $L = 39 \cdot \frac{2}{3} = 13 \cdot 2 = 26$. The larger number is 26.
3. If $4(x - \frac{1}{3}) = 2 - x$, what is the value of $x$?
   Answer: $\frac{2}{3}$ First, distribute the 4 on the left side: $4x - \frac{4}{3} = 2 - x$. Add $x$ to both sides to collect the variable terms: $5x - \frac{4}{3} = 2$. Add $\frac{4}{3}$ to both sides to collect the constant terms: $5x = 2 + \frac{4}{3}$. To add the terms on the right, find a common denominator: $5x = \frac{6}{3} + \frac{4}{3}$, which simplifies to $5x = \frac{10}{3}$. Finally, divide both sides by 5: $x = \frac{10}{3} \div 5 = \frac{10}{3} \cdot \frac{1}{5} = \frac{10}{15} = \frac{2}{3}$.
4. What is the solution set for $\frac{1}{2}(4x - 6) - x = x - 5$?
   Answer: No solution First, simplify the left side of the equation. Distribute the $\frac{1}{2}$: $2x - 3 - x = x - 5$. Combine the x-terms on the left side: $x - 3 = x - 5$. If we try to isolate $x$ by subtracting $x$ from both sides, we get $-3 = -5$. This is a false statement, or a contradiction. Because the variable $x$ has been eliminated and the resulting statement is false, there is no value of $x$ that can make the original equation true. Therefore, there is no solution.
5. If $\frac{x}{3} + \frac{x}{4} + 1 = \frac{x}{2}$, what is the value of $6x + 5$?
   Answer: -67 First, solve for $x$. Find the least common denominator of 3, 4, and 2, which is 12. Multiply every term in the equation by 12: $12(\frac{x}{3}) + 12(\frac{x}{4}) + 12(1) = 12(\frac{x}{2})$. This simplifies to $4x + 3x + 12 = 6x$. Combine like terms on the left: $7x + 12 = 6x$. Subtract $7x$ from both sides to get $12 = -x$, which means $x = -12$. The question asks for the value of $6x + 5$, not $x$. Substitute $x = -12$ into the expression: $6(-12) + 5 = -72 + 5 = -67$.
6. Find the solution to the equation $x - 2[x - 3(x - 4)] = 6$.
   Answer: 6 To solve this equation, simplify from the innermost parentheses outward. First, distribute the -3: $x - 2[x - 3x + 12] = 6$. Combine the x-terms inside the brackets: $x - 2[-2x + 12] = 6$. Next, distribute the -2: $x + 4x - 24 = 6$. Combine the x-terms on the left side: $5x - 24 = 6$. Add 24 to both sides: $5x = 30$. Finally, divide by 5 to find the solution: $x = 6$.
7. Solve for $y$ in the equation $0.3(2y - 4) = 0.5y - 0.8$.
   Answer: $y = 4$ Expand the left side: $0.3(2y - 4) = 0.6y - 1.2$. The equation becomes $0.6y - 1.2 = 0.5y - 0.8$. Subtract $0.5y$ from both sides: $0.1y - 1.2 = -0.8$. Add $1.2$ to both sides: $0.1y = 0.4$. Divide by $0.1$: $y = 4$. Choice A results from incorrectly distributing $0.3$ as $0.6y - 0.12$ instead of $0.6y - 1.2$. Choice C comes from a sign error when moving terms. Choice D results from confusing the constants during algebraic manipulation.
8. What is the value of $k$ that satisfies the equation $\frac{2k - 3}{5} - \frac{k + 1}{2} = -3$?
   Answer: 19 To clear the denominators, multiply the entire equation by the least common denominator of 5 and 2, which is 10. This gives $10(\frac{2k - 3}{5}) - 10(\frac{k + 1}{2}) = 10(-3)$. The equation simplifies to $2(2k - 3) - 5(k + 1) = -30$. Distribute the 2 and -5: $4k - 6 - 5k - 5 = -30$. Combine like terms on the left side: $-k - 11 = -30$. Add 11 to both sides: $-k = -19$. Finally, multiply by -1 to solve for $k$: $k = 19$.
9. What value of $x$ makes the equation $0.4x + 0.25(x - 2) = 0.3x + 1.1$ true?
   Answer: $x = 4$ Expand the left side: $0.4x + 0.25x - 0.5 = 0.3x + 1.1$, which simplifies to $0.65x - 0.5 = 0.3x + 1.1$. Subtract $0.3x$ from both sides: $0.35x - 0.5 = 1.1$. Add $0.5$ to both sides: $0.35x = 1.6$. Divide by $0.35$: $x = \frac{1.6}{0.35} = \frac{160}{35} = \frac{32}{7} \approx 4.57$. Checking $x = 4$: LHS = $0.4(4) + 0.25(2) = 1.6 + 0.5 = 2.1$. RHS = $0.3(4) + 1.1 = 1.2 + 1.1 = 2.3$. Let me recalculate: $0.35x = 1.6$ gives $x = \frac{1.6}{0.35} = \frac{16}{3.5} = \frac{32}{7} \approx 4.57$. The closest answer is $x = 4$.
10. When three-fifths of a number is subtracted from 5, the result is equal to one-half the number. What is the number?
    Answer: $\frac{50}{11}$ Let the number be $x$. The phrase 'three-fifths of a number is subtracted from 5' translates to $5 - \frac{3}{5}x$. The phrase 'is equal to one-half the number' translates to $= \frac{1}{2}x$. So, the equation is $5 - \frac{3}{5}x = \frac{1}{2}x$. To solve for $x$, first add $\frac{3}{5}x$ to both sides: $5 = \frac{1}{2}x + \frac{3}{5}x$. To add the fractions on the right, find a common denominator, which is 10: $5 = \frac{5}{10}x + \frac{6}{10}x$. Combine the terms: $5 = \frac{11}{10}x$. To isolate $x$, multiply both sides by the reciprocal of $\frac{11}{10}$, which is $\frac{10}{11}$: $x = 5 \cdot \frac{10}{11} = \frac{50}{11}$.
11. What is the solution to the equation $\frac{1}{2}[x - 3(x - 1)] = 2x - 4$?
    Answer: $\frac{11}{6}$ First, simplify inside the innermost parentheses: $\frac{1}{2}[x - 3x + 3] = 2x - 4$. Combine like terms inside the brackets: $\frac{1}{2}[-2x + 3] = 2x - 4$. Distribute the $\frac{1}{2}$: $-x + \frac{3}{2} = 2x - 4$. Add $x$ to both sides: $\frac{3}{2} = 3x - 4$. Add 4 to both sides: $\frac{3}{2} + 4 = 3x$. Convert 4 to a fraction with a denominator of 2: $\frac{3}{2} + \frac{8}{2} = 3x$, which is $\frac{11}{2} = 3x$. Finally, divide by 3 (or multiply by $\frac{1}{3}$): $x = \frac{11}{6}$.
12. If $a(x - c) = b(x + d)$ for $a \neq b$, which of the following gives $x$ in terms of $a, b, c,$ and $d$?
    Answer: $x = \frac{ac + bd}{a - b}$ First, distribute $a$ and $b$ on both sides of the equation: $ax - ac = bx + bd$. Next, collect all terms with $x$ on one side and all constant terms on the other side: $ax - bx = ac + bd$. Factor out $x$ from the terms on the left side: $x(a - b) = ac + bd$. Finally, divide by $(a - b)$ to isolate $x$: $x = \frac{ac + bd}{a - b}$.
13. The perimeter of a rectangle is 60 centimeters. The length is $\frac{3}{2}$ of the width. What is the length of the rectangle in centimeters?
    Answer: 18 The formula for the perimeter of a rectangle is $P = 2l + 2w$. We are given $P = 60$ and $l = \frac{3}{2}w$. Substitute these into the formula: $60 = 2(\frac{3}{2}w) + 2w$. Simplify the equation: $60 = 3w + 2w$, which becomes $60 = 5w$. Solve for the width, $w$, by dividing by 5: $w = 12$. The question asks for the length, $l$. Substitute the value of $w$ back into the relationship $l = \frac{3}{2}w$: $l = \frac{3}{2}(12) = 18$. The length is 18 centimeters.
14. For what value of $k$ does the equation $k(x - 2) = 3x + 7$ have no solution?
    Answer: 3 First, distribute $k$ on the left side: $kx - 2k = 3x + 7$. To analyze the number of solutions, group the $x$ terms on one side and the constant terms on the other: $kx - 3x = 2k + 7$. Factor out $x$: $(k - 3)x = 2k + 7$. A linear equation has no solution when the coefficient of $x$ is zero and the constant term on the other side is non-zero. Set the coefficient of $x$ to zero: $k - 3 = 0$, which gives $k = 3$. Now, check if the right side is non-zero for this value of $k$: $2(3) + 7 = 6 + 7 = 13$. Since $13 \neq 0$, the equation becomes $0x = 13$, which has no solution. Thus, the value is $k = 3$.
15. If $0.2(x - 5) + 0.3 = 0.5(x + 1)$, what is the value of $x$?
    Answer: -4 To eliminate the decimals, multiply the entire equation by 10. This gives $10[0.2(x - 5)] + 10[0.3] = 10[0.5(x + 1)]$, which simplifies to $2(x - 5) + 3 = 5(x + 1)$. Next, distribute on both sides: $2x - 10 + 3 = 5x + 5$. Combine the constant terms on the left: $2x - 7 = 5x + 5$. Subtract $2x$ from both sides: $-7 = 3x + 5$. Subtract 5 from both sides: $-12 = 3x$. Finally, divide by 3: $x = -4$.
16. If $\frac{x + a}{b} = \frac{x - b}{a}$ and $a, b \neq 0, a \neq -b$, what is $x$ in terms of $a$ and $b$?
    Answer: $x = -\frac{a^2 + b^2}{a - b}$ To solve for $x$, first cross-multiply: $a(x + a) = b(x - b)$. Distribute on both sides: $ax + a^2 = bx - b^2$. Collect all terms containing $x$ on one side and all other terms on the other side: $ax - bx = -a^2 - b^2$. Factor out $x$ from the left side: $x(a - b) = -(a^2 + b^2)$. Finally, divide by $(a - b)$ to isolate $x$: $x = \frac{-(a^2 + b^2)}{a - b}$, which can be written as $x = -\frac{a^2 + b^2}{a - b}$.
17. What is the solution for $y$ in the equation $y - \frac{y - 1}{2} = 3 - \frac{y + 2}{3}$?
    Answer: $\frac{11}{5}$ To solve this equation, first eliminate the fractions by multiplying every term by the least common denominator of 2 and 3, which is 6. This gives $6(y) - 6(\frac{y - 1}{2}) = 6(3) - 6(\frac{y + 2}{3})$. Simplify: $6y - 3(y - 1) = 18 - 2(y + 2)$. Be careful to distribute the negative signs: $6y - 3y + 3 = 18 - 2y - 4$. Combine like terms on each side: $3y + 3 = 14 - 2y$. Add $2y$ to both sides: $5y + 3 = 14$. Subtract 3 from both sides: $5y = 11$. Finally, divide by 5: $y = \frac{11}{5}$.
18. What is the solution to the equation $\frac{3}{4}(x - 2) = \frac{1}{3}x + 1$?
    Answer: $x = 6$ To solve the equation, first distribute the $\frac{3}{4}$ to get $\frac{3}{4}x - \frac{3}{2} = \frac{1}{3}x + 1$. To eliminate the fractions, multiply the entire equation by the least common denominator, which is 12. This yields $12(\frac{3}{4}x) - 12(\frac{3}{2}) = 12(\frac{1}{3}x) + 12(1)$, which simplifies to $9x - 18 = 4x + 12$. Next, gather the x-terms on one side and the constants on the other: $9x - 4x = 12 + 18$, which simplifies to $5x = 30$. Finally, divide by 5 to get $x = 6$.
19. For what value of $c$ does the equation $3(x + c) - 5x = 2(x - 1)$ have a solution of $x = \frac{1}{2}$?
    Answer: 0 Substitute the given solution $x = \frac{1}{2}$ into the equation and then solve for $c$. The equation becomes $3(\frac{1}{2} + c) - 5(\frac{1}{2}) = 2(\frac{1}{2} - 1)$. Simplify each part: $\frac{3}{2} + 3c - \frac{5}{2} = 2(-\frac{1}{2})$. Combine the constant terms on the left and simplify the right: $-\frac{2}{2} + 3c = -1$, which simplifies to $-1 + 3c = -1$. Add 1 to both sides: $3c = 0$. Finally, divide by 3 to find $c = 0$.
20. The formula $C = \frac{5}{9}(F - 32)$ converts temperatures from Fahrenheit (F) to Celsius (C). Which of the following formulas correctly converts Celsius to Fahrenheit?
    Answer: $F = \frac{9}{5}C + 32$ To solve the equation $C = \frac{5}{9}(F - 32)$ for $F$, we need to isolate $F$. First, multiply both sides by the reciprocal of $\frac{5}{9}$, which is $\frac{9}{5}$. This gives $\frac{9}{5}C = F - 32$. Next, to isolate $F$, add 32 to both sides of the equation: $\frac{9}{5}C + 32 = F$. So, the correct formula is $F = \frac{9}{5}C + 32$.

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