One-Variable Linear Equations - Accuplacer Qas

For what value of $k$ does the equation $7x - 5 = k(x - 1) + 3x$ have no solution?
Answer: $k = 4$ First, simplify the right side of the equation: $kx - k + 3x$ , which can be rewritten as $(k+3)x - k$ . The full equation is $7x - 5 = (k+3)x - k$ . For an equation to have no solution, the coefficients of the variable term must be equal, while the constant terms must be unequal. Equating the coefficients of $x$ gives $7 = k + 3$ , which means $k = 4$ . Now, check if the constants are unequal. The constants are -5 and - $k$ . If $k=4$ , the constants are -5 and -4, which are unequal. Therefore, when $k = 4$ , the equation has no solution.
What is the value of $x$ if $21 = 5(x - 2) + 4x$ ?
Answer: $x = \frac{31}{9}$ First, simplify the right side of the equation. Distribute the 5: $21 = 5x - 10 + 4x$ . Combine the $x$ terms: $21 = 9x - 10$ . Add 10 to both sides: $31 = 9x$ . Finally, divide by 9 to solve for $x$ : $x = \frac{31}{9}$ .
A car rental company charges $25 per day plus $0.15 per mile. If Maria's total bill for a 3-day rental was $127.50, how many miles did she drive?
Answer: 350 miles The total cost equation is: $3(25) + 0.15m = 127.50$ , where $m$ is miles driven. This simplifies to $75 + 0.15m = 127.50$ . Subtracting 75: $0.15m = 52.50$ . Dividing by 0.15: $m = \frac{52.50}{0.15} = 350$ . Choice B results from using the wrong daily rate. Choice C comes from arithmetic errors in division. Choice D results from incorrectly setting up the initial equation.
Find the value of $b$ such that the equation $2(3x - 1) + b = 6x - 2$ has an infinite number of solutions.
Answer: $b = 0$ For an equation to have an infinite number of solutions, both sides of the equation must be identical. First, simplify the left side by distributing the 2: $6x - 2 + b = 6x - 2$ . To make the sides identical, the constant terms must be equal. Here, we must have $-2 + b = -2$ . Adding 2 to both sides gives $b = 0$ .
A plumber charges a flat fee of $65 for a house call, plus $45 per hour for labor. If the final bill was $267.50, for how many hours did the plumber work?
Answer: 4.5 Let $h$ be the number of hours worked. The total cost is the flat fee plus the hourly charge, so the equation is $65 + 45h = 267.50$ . To solve for $h$ , first subtract 65 from both sides: $45h = 202.50$ . Then, divide by 45: $h = \frac{202.50}{45} = 4.5$ . The plumber worked for 4.5 hours.
What is the value of $z$ in the equation $0.5(z - 6) = 0.2z + 3$ ?
Answer: 20 First, distribute 0.5 on the left side: $0.5z - 3 = 0.2z + 3$ . Next, subtract $0.2z$ from both sides to get $0.3z - 3 = 3$ . Then, add 3 to both sides: $0.3z = 6$ . Finally, divide by 0.3: $z = 6 / 0.3 = 20$ .
If four times a number is increased by 3, the result is the same as seven less than six times the number. What is the number?
Answer: 5 Let the number be $x$ . Translate the sentence into an equation. 'Four times a number is increased by 3' is $4x + 3$ . 'Seven less than six times the number' is $6x - 7$ . Set them equal: $4x + 3 = 6x - 7$ . To solve, subtract $4x$ from both sides: $3 = 2x - 7$ . Add 7 to both sides: $10 = 2x$ . Divide by 2: $x = 5$ .
What is the solution to the equation $8 - \frac{3}{2}(2x - 4) = 5$ ?
Answer: $x = 3$ First, distribute $-\frac{3}{2}$ to both terms inside the parentheses: $8 + (-\frac{3}{2})(2x) + (-\frac{3}{2})(-4) = 5$ . This simplifies to $8 - 3x + 6 = 5$ . Combine the constant terms on the left: $14 - 3x = 5$ . Subtract 14 from both sides: $-3x = -9$ . Finally, divide by -3 to get $x = 3$ .
What is the solution to the equation $12 - [3(a - 1) - a] = 5$ ?
Answer: $a = 5$ First, simplify the expression inside the brackets. Distribute the 3: $3a - 3 - a$ , which simplifies to $2a - 3$ . The equation becomes $12 - [2a - 3] = 5$ . Next, distribute the negative sign: $12 - 2a + 3 = 5$ . Combine the constants on the left: $15 - 2a = 5$ . Subtract 15 from both sides: $-2a = -10$ . Finally, divide by -2 to get $a = 5$ .
What is the solution to the equation $\frac{2x - 1}{5} = \frac{x + 3}{4}$ ?
Answer: $x = \frac{19}{3}$ To solve this proportion, you can cross-multiply. Multiply the numerator of the left fraction by the denominator of the right, and set it equal to the numerator of the right fraction multiplied by the denominator of the left: $4(2x - 1) = 5(x + 3)$ . Distribute on both sides: $8x - 4 = 5x + 15$ . Subtract $5x$ from both sides: $3x - 4 = 15$ . Add 4 to both sides: $3x = 19$ . Divide by 3: $x = \frac{19}{3}$ .
The perimeter of a rectangle is 54 centimeters. The length is 3 centimeters more than twice the width. What is the width of the rectangle in centimeters?
Answer: 8 Let $w$ be the width. The length $l$ is $2w + 3$ . The formula for the perimeter is $P = 2(l + w)$ . Substitute the given values: $54 = 2((2w + 3) + w)$ . Simplify inside the parentheses: $54 = 2(3w + 3)$ . Distribute the 2: $54 = 6w + 6$ . Subtract 6 from both sides: $48 = 6w$ . Divide by 6 to find the width: $w = 8$ .
What is the solution to the equation $\frac{1}{2}x - 3 = \frac{1}{4}x + 1$ ?
Answer: $x = 16$ To solve this equation, first eliminate the fractions by multiplying every term by the least common denominator, which is 4. This gives: $4(\frac{1}{2}x) - 4(3) = 4(\frac{1}{4}x) + 4(1)$ , which simplifies to $2x - 12 = x + 4$ . Next, subtract $x$ from both sides to get $x - 12 = 4$ . Finally, add 12 to both sides to get $x = 16$ .
Which of the following best describes the solution to the equation $4(y - 1) - 2y = 2(y + 3)$ ?
Answer: No solution First, simplify both sides of the equation. On the left, distribute the 4: $4y - 4 - 2y$ , which combines to $2y - 4$ . On the right, distribute the 2: $2y + 6$ . The equation becomes $2y - 4 = 2y + 6$ . If you subtract $2y$ from both sides, you get $-4 = 6$ , which is a false statement. Because the variables cancel out and the resulting statement is false, there is no value of $y$ that can make the equation true. Therefore, there is no solution.
The sum of three consecutive odd integers is 87. What is the value of the largest of these integers?
Answer: 31 Let the three consecutive odd integers be represented by $n$ , $n + 2$ , and $n + 4$ . Their sum is 87, so the equation is $n + (n + 2) + (n + 4) = 87$ . Combining like terms gives $3n + 6 = 87$ . Subtracting 6 from both sides gives $3n = 81$ . Dividing by 3 gives $n = 27$ . This is the smallest integer. The question asks for the largest integer, which is $n + 4$ . So, the largest integer is $27 + 4 = 31$ .
Which of the following best describes the solution to the equation $-(w + 5) + 3w = 2(w - 3) + 1$ ?
Answer: All real numbers First, simplify both sides of the equation. On the left, distribute the negative sign: $-w - 5 + 3w$ , which combines to $2w - 5$ . On the right, distribute the 2: $2w - 6 + 1$ , which combines to $2w - 5$ . The equation becomes $2w - 5 = 2w - 5$ . This statement is always true, regardless of the value of $w$ . Because the two sides are identical, any real number substituted for $w$ will result in a true statement. Therefore, the solution is all real numbers.
If $\frac{3}{5}y + 2 = 11$ , what is the value of $y + 5$ ?
Answer: 20 First, solve for $y$ . Subtract 2 from both sides of the equation to get $\frac{3}{5}y = 9$ . To isolate $y$ , multiply both sides by the reciprocal of $\frac{3}{5}$ , which is $\frac{5}{3}$ . This gives $y = 9 \times \frac{5}{3} = 15$ . The question asks for the value of $y + 5$ , so substitute $y=15$ into the expression: $15 + 5 = 20$ .
A phone plan charges a monthly fee plus a cost per minute for calls. If 200 minutes costs $45 total and 350 minutes costs $67.50 total, what would be the total cost for 500 minutes?
Answer: $90.00 Let $f$ be the monthly fee and $c$ be the cost per minute. We have: $f + 200c = 45$ and $f + 350c = 67.50$ . Subtracting: $150c = 22.50$ , so $c = 0.15$ . Substituting: $f + 200(0.15) = 45$ , so $f + 30 = 45$ and $f = 15$ . For 500 minutes: $15 + 500(0.15) = 15 + 75 = 90$ . Choice A results from using the wrong per-minute rate. Choice C comes from arithmetic errors in the final calculation. Choice D results from incorrectly calculating the monthly fee.
A plumber charges a flat service fee plus an hourly rate. If the total cost for a 3-hour job is $285 and the total cost for a 5-hour job is $425, what would be the total cost for a 7-hour job?
Answer: $565 Let the service fee be $s$ and hourly rate be $h$ . We have the system: $s + 3h = 285$ and $s + 5h = 425$ . Subtracting the first equation from the second: $2h = 140$ , so $h = 70$ . Substituting back: $s + 3(70) = 285$ , so $s = 75$ . For a 7-hour job: $75 + 7(70) = 75 + 490 = 565$ . Choice B assumes the hourly rate is $75 instead of $70. Choice C results from using the wrong service fee. Choice D comes from arithmetic errors in the final calculation.
What is the solution to the equation $9x - 5 = 3(x + 4)$ ?
Answer: $x = \frac{17}{6}$ First, distribute the 3 on the right side: $9x - 5 = 3x + 12$ . Next, subtract $3x$ from both sides to get $6x - 5 = 12$ . Then, add 5 to both sides to get $6x = 17$ . Finally, divide by 6 to isolate $x$ : $x = \frac{17}{6}$ .
What is the solution to the equation $5(n - 2) - (n + 4) = 3n - 8$ ?
Answer: $n = 6$ First, distribute on the left side: $5n - 10 - n - 4 = 3n - 8$ . Be careful to distribute the negative sign to both terms in the parenthesis. Combine like terms on the left: $4n - 14 = 3n - 8$ . Subtract $3n$ from both sides: $n - 14 = -8$ . Add 14 to both sides: $n = 6$ .

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