Solve Linear Equations - Praxis Core Math

1. If $\frac{4x+1}{3} - 2 = x$, what is the value of $x$?
   Answer: 5 To solve this equation, first eliminate the fraction by multiplying every term by 3: $3(\frac{4x+1}{3}) - 3(2) = 3(x)$. This simplifies to $4x+1 - 6 = 3x$. Combine the constant terms on the left: $4x - 5 = 3x$. To solve for $x$, subtract $3x$ from both sides: $x - 5 = 0$. Finally, add 5 to both sides: $x = 5$.
2. If $\frac{1}{2}x + \frac{1}{3}x + \frac{1}{4}x = 26$, what is the value of $x$?
   Answer: 24 To solve for $x$, first find a common denominator for the fractions, which is 12. Rewrite the equation with the common denominator: $\frac{6}{12}x + \frac{4}{12}x + \frac{3}{12}x = 26$. Combine the terms on the left: $\frac{13}{12}x = 26$. To isolate $x$, multiply both sides by the reciprocal of $\frac{13}{12}$, which is $\frac{12}{13}$: $x = 26 \times \frac{12}{13}$. Since $26/13 = 2$, the equation becomes $x = 2 \times 12 = 24$.
3. What is the value of $y$ that satisfies the equation $4(y - 1) = 2(2y + 3)$?
   Answer: There is no solution. Begin by distributing on both sides of the equation: $4y - 4 = 4y + 6$. Next, attempt to isolate the variable $y$. Subtract $4y$ from both sides: $-4 = 6$. This results in a statement that is false. A false statement like this indicates that there is no value of $y$ that can make the original equation true. Therefore, the equation has no solution.
4. For what value of $k$ does the equation $3(x + 2) = 3x + k$ have infinitely many solutions?
   Answer: 6 An equation has infinitely many solutions if it is an identity, meaning the expressions on both sides are equivalent for all values of $x$. First, distribute the 3 on the left side of the equation: $3x + 6 = 3x + k$. For the two sides to be identical, the constant terms must be equal. Therefore, $k$ must be equal to 6.
5. A cell phone plan costs $30 per month and includes 2 gigabytes (GB) of data. Each additional gigabyte of data costs $10. If a customer's bill was $75 for one month, not including taxes, how many total gigabytes of data did the customer use?
   Answer: 6.5 GB Let $g$ be the number of additional gigabytes used. The cost for additional data is $10g$. The total bill is the base cost plus the additional data cost: $30 + 10g = 75$. To find $g$, first subtract 30 from both sides: $10g = 45$. Then divide by 10: $g = 4.5$. This represents the additional gigabytes. The question asks for the total gigabytes used, which is the included 2 GB plus the additional 4.5 GB. Total data = $2 + 4.5 = 6.5$ GB.
6. Two angles are supplementary. One angle measures $(4x + 20)$ degrees and the other measures $(2x + 10)$ degrees. What is the measure of the larger angle?
   Answer: 120 degrees Supplementary angles add up to 180 degrees. So, we set up the equation: $(4x + 20) + (2x + 10) = 180$. Combine like terms: $6x + 30 = 180$. Subtract 30 from both sides: $6x = 150$. Divide by 6: $x = 25$. This is the value of $x$, not the angle measure. Substitute $x = 25$ into both expressions to find the angles: $4(25) + 20 = 100 + 20 = 120$ degrees, and $2(25) + 10 = 50 + 10 = 60$ degrees. The larger of the two angles is 120 degrees.
7. What is the solution to the equation $\frac{2}{3}y + 5 = \frac{1}{2}y - 1$?
   Answer: -36 To eliminate the fractions, multiply the entire equation by the least common denominator of 3 and 2, which is 6. This gives: $6(\frac{2}{3}y) + 6(5) = 6(\frac{1}{2}y) - 6(1)$. Simplifying, we get $4y + 30 = 3y - 6$. To solve for $y$, subtract $3y$ from both sides: $y + 30 = -6$. Then, subtract 30 from both sides: $y = -36$.
8. At a community bake sale, pies sell for $15 and cookies sell for $2. A total of 50 items were sold, generating a revenue of $360. How many pies were sold?
   Answer: 20 Let $p$ be the number of pies sold. Since a total of 50 items were sold, the number of cookies sold is $50 - p$. The total revenue is the sum of the revenue from pies and cookies: $15p + 2(50 - p) = 360$. Distribute the 2: $15p + 100 - 2p = 360$. Combine like terms: $13p + 100 = 360$. Subtract 100 from both sides: $13p = 260$. Divide by 13: $p = 20$. Thus, 20 pies were sold.
9. If $7k - 11 = 3k + 5$, what is the value of the expression $2k + 1$?
   Answer: 9 This is a two-step problem. First, solve the linear equation for $k$. Subtract (3k) from both sides: $4k - 11 = 5$. Add 11 to both sides: $4k = 16$. Divide by 4: $k = 4$. Second, substitute this value of $k$ into the expression $2k + 1$. This gives $2(4) + 1 = 8 + 1 = 9$. The value of the expression is 9.
10. What is the solution to the equation $\frac{z-4}{3} = \frac{z+2}{5}$?
    Answer: 13 To solve this proportion, cross-multiply: $5(z-4) = 3(z+2)$. Distribute on both sides: $5z - 20 = 3z + 6$. Next, get the variable terms on one side by subtracting $3z$ from both sides: $2z - 20 = 6$. Then, isolate the variable term by adding 20 to both sides: $2z = 26$. Finally, divide by 2: $z = 13$.
11. The perimeter of a triangle is 50 inches. The first side is 6 inches longer than the second side. The third side is twice the length of the second side. What is the length of the longest side?
    Answer: 22 inches Let the length of the second side be $x$. Then the first side is $x+6$, and the third side is $2x$. The perimeter is the sum of the sides: $x + (x+6) + 2x = 50$. Combine like terms: $4x + 6 = 50$. Subtract 6 from both sides: $4x = 44$. Divide by 4: $x = 11$. This is the length of the second side. The lengths of the sides are: second side = 11 inches, first side = $11+6=17$ inches, and third side = $2(11)=22$ inches. The longest side is 22 inches.
12. Half of a number increased by 10 is equal to five less than the number. What is the number?
    Answer: 30 Let the number be $n$. Translating the sentence into a mathematical equation gives: $\frac{1}{2}n + 10 = n - 5$. To solve for $n$, first gather the $n$ terms on one side. Subtract $\frac{1}{2}n$ from both sides: $10 = \frac{1}{2}n - 5$. Next, add 5 to both sides: $15 = \frac{1}{2}n$. Finally, multiply both sides by 2 to isolate $n$: $30 = n$. The number is 30.
13. If a number is doubled and then decreased by 7, the result is the same as if the number were increased by 5. What is the number?
    Answer: 12 Let the number be $x$. Translating the words into an equation gives: $2x - 7 = x + 5$. To solve for $x$, subtract $x$ from both sides: $x - 7 = 5$. Then, add 7 to both sides: $x = 12$. The number is 12.
14. The equation for converting temperature from Celsius ($C$) to Fahrenheit ($F$) is $F = \frac{9}{5}C + 32$. Which of the following equations correctly expresses $C$ in terms of $F$?
    Answer: $C = \frac{5}{9}(F - 32)$ To solve the equation $F = \frac{9}{5}C + 32$ for $C$, we must isolate $C$. First, subtract 32 from both sides: $F - 32 = \frac{9}{5}C$. Next, to get $C$ by itself, multiply both sides by the reciprocal of $\frac{9}{5}$, which is $\frac{5}{9}$. This gives $\frac{5}{9}(F - 32) = C$. Thus, $C = \frac{5}{9}(F - 32)$.
15. The sum of three consecutive odd integers is 141. What is the largest of these three integers?
    Answer: 49 Let the three consecutive odd integers be $n$, $n + 2$, and $n + 4$. Their sum is $n + (n + 2) + (n + 4) = 141$. Combine like terms: $3n + 6 = 141$. Subtract 6 from both sides: $3n = 135$. Divide by 3: $n = 45$. This is the smallest of the three integers. The integers are 45, 47, and 49. The question asks for the largest integer, which is 49.
16. The equation $A = P(1+rt)$ gives the total amount $A$ in an account with principal $P$, annual interest rate $r$, and time in years $t$. If an account grows to $5,250 in 10 years with an initial principal of $3,000, what is the value of $r$?
    Answer: 0.075 Substitute the given values into the equation: $5250 = 3000(1 + r \cdot 10)$. First, divide both sides by 3000: $\frac{5250}{3000} = 1 + 10r$. The fraction simplifies to $1.75$. So, $1.75 = 1 + 10r$. Next, subtract 1 from both sides: $0.75 = 10r$. Finally, divide by 10 to solve for $r$: $r = \frac{0.75}{10} = 0.075$.
17. An investment of $1,200 earns simple interest. After 5 years, the total value of the investment is $1,500. Using the simple interest formula $I = Prt$, where $I$ is interest, $P$ is principal, $r$ is the annual rate, and $t$ is time in years, what was the annual interest rate?
    Answer: 5% First, calculate the total interest earned ($I$) by subtracting the principal ($P$) from the final amount: $I = \$1,500 - \$1,200 = \$300$. Now, use the formula $I = Prt$ with $I=300$, $P=1200$, and $t=5$. The equation is $300 = 1200 \times r \times 5$. Simplify the right side: $300 = 6000r$. To find the rate $r$, divide both sides by 6000: $r = \frac{300}{6000} = \frac{3}{60} = \frac{1}{20} = 0.05$. To express this as a percentage, multiply by 100. The annual interest rate is 5%.
18. If $10 - 2(3x - 1) = 4x$, what is the value of $x$?
    Answer: 1.2 First, distribute the -2 to both terms inside the parentheses: $10 - 6x + 2 = 4x$. Be careful with the signs; $-2 \times -1 = +2$. Combine the constant terms on the left side: $12 - 6x = 4x$. To isolate the $x$ term, add $6x$ to both sides: $12 = 10x$. Finally, divide both sides by 10: $x = \frac{12}{10} = 1.2$.
19. What is the solution to the equation $4[x - 2(x-3)] = 8$?
    Answer: 4 To solve this equation, work from the inside out. First, distribute the -2 inside the brackets: $4[x - 2x + 6] = 8$. Combine the $x$ terms inside the brackets: $4[-x + 6] = 8$. Next, distribute the 4: $-4x + 24 = 8$. Subtract 24 from both sides: $-4x = -16$. Finally, divide by -4: $x = 4$. Alternatively, one could divide both sides by 4 first: $x - 2(x-3) = 2$, which leads to the same result.
20. Maria is twice as old as her brother, Sam. In 6 years, the sum of their ages will be 39. How old is Sam now?
    Answer: 9 Let Sam's current age be $S$. Then Maria's current age is $2S$. In 6 years, Sam's age will be $S+6$ and Maria's age will be $2S+6$. The sum of their ages in 6 years will be 39, so $(S+6) + (2S+6) = 39$. Combine like terms: $3S + 12 = 39$. Subtract 12 from both sides: $3S = 27$. Divide by 3: $S = 9$. Sam is currently 9 years old.

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