Examkin

Linear Equations in One Variable

67 questions

All types67Multiple choice35Numeric32
Question 41Question →

If 12(4x−6)−2(x+5)=3x−1\frac{1}{2}(4x - 6) - 2(x + 5) = 3x - 1, what is the value of xx?

Show answer & explanation

Answer: −4-4

Answer

−4-4
The correct answer is −4-4. Distributing the terms on the left side of the equation 12(4x−6)−2(x+5)=3x−1\frac{1}{2}(4x - 6) - 2(x + 5) = 3x - 1 gives 2x−3−2x−10=3x−12x - 3 - 2x - 10 = 3x - 1. Simplifying the left side yields −13=3x−1-13 = 3x - 1. Adding 11 to both sides results in −12=3x-12 = 3x. Finally, dividing both sides by 33 gives x=−4x = -4.

Step-by-Step Solution

1
Distribute the constants outside the parentheses on the left side of the equation.
2x−3−2x−10=3x−12x - 3 - 2x - 10 = 3x - 1
Distributing 12\frac{1}{2} to (4x−6)(4x - 6) yields 2x−32x - 3, and distributing −2-2 to (x+5)(x + 5) yields −2x−10-2x - 10.
2
Combine like terms on the left side of the equation.
−13=3x−1-13 = 3x - 1
Combining 2x2x and −2x-2x results in 00, and combining −3-3 and −10-10 results in −13-13.
3
Isolate the variable xx by adding 11 to both sides and dividing by 33.
x=−4x = -4
Adding 11 to both sides gives −12=3x-12 = 3x. Dividing both sides by 33 yields x=−4x = -4.

Key Concept

Solving linear equations in one variable by distributing constants and combining like terms.
Question 42Question →

If 12−2(w−1)=412 - 2(w - 1) = 4, what is the value of ww?

Show answer & explanation

Answer: 5

Answer

5
The correct answer is 5. We solve the equation by first distributing the coefficient −2-2 to (w−1)(w - 1), which yields 12−2w+2=412 - 2w + 2 = 4. Combining the constant terms gives 14−2w=414 - 2w = 4. Subtracting 14 from both sides results in −2w=−10-2w = -10. Finally, dividing by −2-2 isolates the variable, giving w=5w = 5.

Step-by-Step Solution

1
Distribute the negative coefficient −2-2 to the terms inside the parentheses.
12−2w+2=412 - 2w + 2 = 4
To eliminate the parentheses and simplify the expression.
2
Combine the constant terms on the left side of the equation.
14−2w=414 - 2w = 4
To group like terms together.
3
Subtract 14 from both sides of the equation.
−2w=−10-2w = -10
To isolate the variable term on one side of the equation.
4
Divide both sides of the equation by −2-2.
w=5w = 5
To find the value of ww.

Key Concept

Solving linear equations in one variable using distributive properties and basic operations.
Question 43Question →
In the equation below, pp and qq are constants.
2p(6x−9)−3q(4x+2)=−2x−9\frac{2}{p}(6x - 9) - \frac{3}{q}(4x + 2) = -2x - 9
If the equation has infinitely many solutions for xx, what is the value of p+qp + q?
Show answer & explanation

Answer: 5

Answer

The value of p+qp+q is 55.
For the equation to have infinitely many solutions for xx, the coefficients of xx on both sides must be equal, and the constant terms on both sides must also be equal. Expanding the left side of the equation gives (12p−12q)x−(18p+6q)=−2x−9\left(\frac{12}{p} - \frac{12}{q}\right)x - \left(\frac{18}{p} + \frac{6}{q}\right) = -2x - 9. Equating the coefficients of xx yields 12p−12q=−2\frac{12}{p} - \frac{12}{q} = -2, which simplifies to 6p−6q=−1\frac{6}{p} - \frac{6}{q} = -1. Equating the constant terms yields −18p−6q=−9-\frac{18}{p} - \frac{6}{q} = -9, which simplifies to 18p+6q=9\frac{18}{p} + \frac{6}{q} = 9. Adding these two simplified equations eliminates the term with qq, giving 24p=8\frac{24}{p} = 8, which means p=3p = 3. Substituting p=3p = 3 back into the first equation yields 2−6q=−12 - \frac{6}{q} = -1, which simplifies to 6q=3\frac{6}{q} = 3, meaning q=2q = 2. Thus, the value of p+qp + q is 3+2=53 + 2 = 5.

Step-by-Step Solution

1
Expand and group the terms on the left side of the equation.
(12p−12q)x−(18p+6q)=−2x−9\left(\frac{12}{p} - \frac{12}{q}\right)x - \left(\frac{18}{p} + \frac{6}{q}\right) = -2x - 9
To analyze the linear equation, we must group the coefficients of the variable xx and the constant terms.
2
Set up a system of equations for pp and qq using the condition for infinitely many solutions.
12p−12q=−2\frac{12}{p} - \frac{12}{q} = -2 and −18p−6q=−9-\frac{18}{p} - \frac{6}{q} = -9
A linear equation in the form Ax+B=Cx+DAx + B = Cx + D has infinitely many solutions if and only if A=CA = C and B=DB = D.
3
Simplify the system of equations.
6p−6q=−1\frac{6}{p} - \frac{6}{q} = -1 (Equation 1) and 18p+6q=9\frac{18}{p} + \frac{6}{q} = 9 (Equation 2)
Dividing Equation 1 by 22 and Equation 2 by −1-1 simplifies the coefficients, making the system easier to solve.
4
Solve for pp by adding Equation 1 and Equation 2.
24p=8  ⟹  p=3\frac{24}{p} = 8 \implies p = 3
Adding the two equations eliminates the term containing qq, allowing us to solve directly for pp.
5
Substitute p=3p = 3 back into Equation 1 to solve for qq.
63−6q=−1  ⟹  2−6q=−1  ⟹  6q=3  ⟹  q=2\frac{6}{3} - \frac{6}{q} = -1 \implies 2 - \frac{6}{q} = -1 \implies \frac{6}{q} = 3 \implies q = 2
Using the value of pp allows us to isolate and solve for qq.
6
Calculate the sum of pp and qq.
p+q=3+2=5p + q = 3 + 2 = 5
The question asks for the value of p+qp + q.

Key Concept

Conditions for a linear equation in one variable to have infinitely many solutions, and solving systems of literal equations.
Question 44Question →

If 2x+7=152x + 7 = 15, what is the value of 4x−34x - 3?

Show answer & explanation

Answer: 13

Answer

13
To find the value of the expression, first solve the linear equation for the variable. Subtracting 7 from both sides of the equation yields 2x=82x = 8. Dividing both sides of the equation by 2 yields x=4x = 4. Substituting this value back into the expression yields 4(4)−3=16−3=134(4) - 3 = 16 - 3 = 13. Alternatively, since 2x=82x = 8, multiplying both sides by 2 gives 4x=164x = 16. Subtracting 3 from both sides of this relation yields 4x−3=16−3=134x - 3 = 16 - 3 = 13.

Step-by-Step Solution

1
Subtract 7 from both sides of the equation
2x=82x = 8
To isolate the variable term on one side of the equation
2
Divide both sides by 2
x=4x = 4
To solve for the variable
3
Substitute the value of the variable into the target expression
1313
To find the final evaluated value requested by the question

Key Concept

Solving a linear equation in one variable and evaluating an expression
Question 45Question →

A chemist mixes two saline solutions. Solution A is 12%12\% salt by mass, and Solution B is 30%30\% salt by mass. The mass of Solution B used in the mixture is 20 grams more than 13\frac{1}{3} of the mass of Solution A used. If the resulting mixture is 18%18\% salt by mass, what is the mass, in grams, of Solution A used?

Show answer & explanation

Answer: 120

Answer

The mass of Solution A used is 120 grams.
The correct answer is 120. By setting the mass of Solution A to xx, the mass of Solution B is 13x+20\frac{1}{3}x + 20. Equating the total salt content from both individual solutions to the total salt content of the final mixture gives the linear equation 0.12x+0.30(13x+20)=0.18(x+13x+20)0.12x + 0.30\left(\frac{1}{3}x + 20\right) = 0.18\left(x + \frac{1}{3}x + 20\right). Simplifying both sides yields 0.22x+6=0.24x+3.60.22x + 6 = 0.24x + 3.6. Solving this equation gives 0.02x=2.40.02x = 2.4, which simplifies to x=120x = 120.

Step-by-Step Solution

1
Define the variable xx as the mass, in grams, of Solution A used in the mixture, and express the mass of Solution B in terms of xx.
Mass of Solution A = xx grams; Mass of Solution B = 13x+20\frac{1}{3}x + 20 grams.
To set up expressions representing the mass of each solution in the mixture.
2
Calculate the mass of salt contributed by each solution and write an expression for the total mass of salt.
Salt from Solution A = 0.12x0.12x grams; Salt from Solution B = 0.30(13x+20)=0.10x+60.30\left(\frac{1}{3}x + 20\right) = 0.10x + 6 grams; Total salt = 0.22x+60.22x + 6 grams.
To find the total amount of salt before mixing.
3
Express the total mass of the mixture and the total salt content using the final mixture's percentage.
Total mass of mixture = 43x+20\frac{4}{3}x + 20 grams; Total salt in final mixture = 0.18(43x+20)=0.24x+3.60.18\left(\frac{4}{3}x + 20\right) = 0.24x + 3.6 grams.
To write the total salt content in terms of the final mixture's concentration.
4
Equate the two expressions for the total mass of salt and solve the linear equation for xx.
0.22x+6=0.24x+3.6  ⟹  2.4=0.02x  ⟹  x=1200.22x + 6 = 0.24x + 3.6 \implies 2.4 = 0.02x \implies x = 120.
To find the mass of Solution A that satisfies the mixture conditions.

Key Concept

Setting up and solving a linear equation in one variable from a mixture word problem.
Question 46Question →

If 56(y−2)−13(2y−5)=32\frac{5}{6}(y - 2) - \frac{1}{3}(2y - 5) = \frac{3}{2}, what is the value of 4y4y?

Show answer & explanation

Answer: 36

Answer

The correct answer is 36.
To solve the equation 56(y−2)−13(2y−5)=32\frac{5}{6}(y - 2) - \frac{1}{3}(2y - 5) = \frac{3}{2}, we first clear the fractions by multiplying the entire equation by the least common denominator, 6, which yields 5(y−2)−2(2y−5)=95(y - 2) - 2(2y - 5) = 9. Distributing the terms on the left side gives 5y−10−4y+10=95y - 10 - 4y + 10 = 9. Combining like terms simplifies this to y=9y = 9. The question asks for the value of 4y4y, so we multiply 99 by 44 to get the final answer of 3636.

Step-by-Step Solution

1
Multiply both sides of the equation by the least common denominator, which is 6.
5(y−2)−2(2y−5)=95(y - 2) - 2(2y - 5) = 9
Multiplying by 6 eliminates the fractions, simplifying the equation.
2
Distribute the coefficients across the parentheses.
5y−10−4y+10=95y - 10 - 4y + 10 = 9
Expanding the terms allows like terms to be combined in the next step.
3
Combine the variable terms and constant terms on the left side of the equation.
y=9y = 9
Combining 5y−4y5y - 4y yields yy, and combining −10+10-10 + 10 yields 00, isolating the variable.
4
Multiply the value of yy by 4.
4y=364y = 36
The question asks for the value of 4y4y rather than just yy.

Key Concept

Solving a linear equation in one variable by clearing fractions and combining like terms.
Question 47Question →

In the equation below, aa is a constant.

4(x−2)−a(3−2x)=8x−54(x - 2) - a(3 - 2x) = 8x - 5

If the equation has no solution for xx, what is the value of aa?

Show answer & explanation

Answer: 2

Answer

2
For the linear equation to have no solution, the coefficients of the xx terms on both sides of the equation must be equal, while the constant terms must be different. First, expand the left side of the equation: 4(x−2)−a(3−2x)=4x−8−3a+2ax4(x - 2) - a(3 - 2x) = 4x - 8 - 3a + 2ax. Grouping the xx terms and constant terms gives (4+2a)x−(8+3a)=8x−5(4 + 2a)x - (8 + 3a) = 8x - 5. Setting the coefficient of xx on the left side equal to the coefficient of xx on the right side gives 4+2a=84 + 2a = 8. Solving for aa yields 2a=42a = 4, which simplifies to a=2a = 2. Substituting a=2a = 2 back into the equation yields 8x−14=8x−58x - 14 = 8x - 5, or −14=−5-14 = -5, which is a false statement with no solution. Therefore, the value of aa is 22.

Step-by-Step Solution

1
Expand and simplify the left side of the equation.
The left side expands to 4x−8−3a+2ax4x - 8 - 3a + 2ax. Grouping the terms by the variable xx gives (4+2a)x−(8+3a)(4 + 2a)x - (8 + 3a).
To find when the linear equation has no solution, we need to rewrite it in the standard form px+q=rx+spx + q = rx + s by distributing terms and grouping like terms.
2
Equate the coefficients of the xx terms from both sides of the equation.
4+2a=84 + 2a = 8
A linear equation of the form px+q=rx+spx + q = rx + s has no solution if the coefficients of the variable are equal (p=rp = r) but the constants are not (q≠sq \neq s).
3
Solve for the constant aa.
2a=4  ⟹  a=22a = 4 \implies a = 2
Subtracting 44 from both sides of 4+2a=84 + 2a = 8 isolates the variable term, and dividing by 22 gives the value of aa.
4
Verify that the constants are different when a=2a = 2.
Substituting a=2a = 2 back into the constant terms yields a left-side constant of −(8+3(2))=−14-(8 + 3(2)) = -14 and a right-side constant of −5-5. Since −14≠−5-14 \neq -5, the equation has no solution.
If the constant terms were also equal, the equation would have infinitely many solutions instead of no solution.

Key Concept

Linear Equations in One Variable (No Solution Case)
Question 48Question →

If the equation 34(8x−12)−56(6x−18)=−12(4x−8)\frac{3}{4}(8x - 12) - \frac{5}{6}(6x - 18) = -\frac{1}{2}(4x - 8) is true, what is the value of xx?

Show answer & explanation

Answer: −23-\frac{2}{3}

Answer

−23-\frac{2}{3}
The correct answer is found by distributing the coefficients across each set of parentheses, combining like terms on the left side to get x+6=−2x+4x + 6 = -2x + 4, adding 2x2x to both sides to get 3x+6=43x + 6 = 4, subtracting 66 from both sides to get 3x=−23x = -2, and finally dividing by 33 to obtain −23-\frac{2}{3}.

Step-by-Step Solution

1
Distribute the coefficients to the terms inside the parentheses on both sides of the equation.
6x−9−5x+15=−2x+46x - 9 - 5x + 15 = -2x + 4
To eliminate the parentheses and prepare the equation for combining like terms.
2
Combine the constant and variable terms on the left side of the equation.
x+6=−2x+4x + 6 = -2x + 4
To simplify the left side of the equation.
3
Add 2x2x to both sides of the equation.
3x+6=43x + 6 = 4
To group all terms containing the variable xx on the left side.
4
Subtract 66 from both sides, then divide by 33 to solve for xx.
3x=−23x = -2, which gives x=−23x = -\frac{2}{3}
To isolate xx and find its value.

Key Concept

Linear Equations in One Variable
Question 49Question →

If 3(2.5x−4)−1.5(x−2)=0.8(5x+10)3(2.5x - 4) - 1.5(x - 2) = 0.8(5x + 10), what is the value of xx?

Show answer & explanation

Answer: 8.5

Answer

8.5
By distributing and simplifying terms, the linear equation simplifies to 2x=172x = 17, which yields x=8.5x = 8.5.

Step-by-Step Solution

1
Distribute the constants on both sides of the equation.
7.5x−12−1.5x+3=4x+87.5x - 12 - 1.5x + 3 = 4x + 8
To simplify the expressions.
2
Combine like terms on the left side.
6x−9=4x+86x - 9 = 4x + 8
To simplify the left side of the equation.
3
Subtract 4x4x from both sides of the equation.
2x−9=82x - 9 = 8
To group variable terms on one side.
4
Add 9 to both sides of the equation.
2x=172x = 17
To isolate the term with the variable.
5
Divide both sides by 2.
x=8.5x = 8.5
To find the final value of the variable.

Key Concept

Linear Equations in One Variable

Alternative Method

Multiply the entire equation by 10 to clear all decimal values: 30(2.5x−4)−15(x−2)=8(5x+10)30(2.5x - 4) - 15(x - 2) = 8(5x + 10). Expanding this yields 75x−120−15x+30=40x+8075x - 120 - 15x + 30 = 40x + 80, which simplifies to 60x−90=40x+8060x - 90 = 40x + 80. Subtracting 40x40x and adding 90 to both sides gives 20x=17020x = 170, resulting in x=8.5x = 8.5.
Estimated Time:1m 30s
Question 50Question →

A company manufactures custom travel mugs. The daily cost CC, in dollars, to manufacture nn mugs is given by the formula C=1.25n+250C = 1.25n + 250. The company sells each mug for 4.50.Tomakeaprofitofexactly4.50. To make a profit of exactly 400$ in a day, how many mugs must the company manufacture?

Show answer & explanation

Answer: 200

Answer

The company must manufacture 200 mugs to make a profit of exactly $400 in a day.
Subtracting the cost expression from the revenue expression gives the profit equation: 4.50n−(1.25n+250)=4004.50n - (1.25n + 250) = 400. Distributing the negative sign results in 4.50n−1.25n−250=4004.50n - 1.25n - 250 = 400, which simplifies to 3.25n−250=4003.25n - 250 = 400. Adding 250 to both sides gives 3.25n=6503.25n = 650. Dividing both sides by 3.25 yields n=200n = 200.

Step-by-Step Solution

1
Set up the profit equation
4.50n−(1.25n+250)=4004.50n - (1.25n + 250) = 400
Profit is calculated as total revenue (4.50n4.50n) minus total cost (1.25n+2501.25n + 250).
2
Distribute the negative sign and simplify the equation
3.25n−250=4003.25n - 250 = 400
Subtracting the entire cost expression requires distributing the subtraction to both 1.25n1.25n and 250250, yielding 4.50n−1.25n−2504.50n - 1.25n - 250.
3
Isolate the variable term by adding 250 to both sides
3.25n=6503.25n = 650
Adding 250 to both sides of the equation isolates the term containing nn.
4
Solve for nn by dividing both sides by 3.25
n=200n = 200
Dividing 650 by 3.25 gives the number of mugs needed to achieve the target profit.

Key Concept

Solving linear equations in one variable derived from a real-world context.
Question 51Question →

For what value of the constant kk does the equation 12(2kx−6)−23(9−3x)=5x+4\frac{1}{2}(2kx - 6) - \frac{2}{3}(9 - 3x) = 5x + 4 have no solution for xx?

Show answer & explanation

Answer: 3

Answer

The correct value of kk is 33.
To find the value of kk for which the equation has no solution, we first distribute the coefficients on the left side of the equation to get kx−3−6+2x=5x+4kx - 3 - 6 + 2x = 5x + 4. Combining the like terms on the left side gives (k+2)x−9=5x+4(k + 2)x - 9 = 5x + 4. Moving all the xx terms to one side yields (k−3)x=13(k - 3)x = 13. For a linear equation of the form Ax=BAx = B to have no solution, the coefficient of the variable AA must equal 00 while the constant BB must be non-zero. Setting k−3=0k - 3 = 0 gives the correct value of 33.

Step-by-Step Solution

1
Distribute the fractional coefficients on the left side of the equation.
The term 12(2kx−6)\frac{1}{2}(2kx - 6) simplifies to kx−3kx - 3, and the term −23(9−3x)-\frac{2}{3}(9 - 3x) simplifies to −6+2x-6 + 2x.
Applying the distributive property simplifies the parenthetical expressions.
2
Substitute the simplified expressions back into the equation and group like terms.
The equation becomes kx−3−6+2x=5x+4kx - 3 - 6 + 2x = 5x + 4, which simplifies to (k+2)x−9=5x+4(k + 2)x - 9 = 5x + 4.
Grouping terms allows us to isolate the variable.
3
Isolate the terms containing xx on one side of the equation.
Subtract 5x5x and add 99 to both sides, yielding (k−3)x=13(k - 3)x = 13.
This puts the equation into the standard linear form Ax=BAx = B.
4
Set the coefficient of xx to 00 to find the value of kk that yields no solution.
Setting k−3=0k - 3 = 0 gives k=3k = 3. Since 13≠013 \neq 0, the equation 0x=130x = 13 has no solution.
A linear equation of the form Ax=BAx = B has no solution if and only if A=0A = 0 and B≠0B \neq 0.

Key Concept

Determining the parameter value for which a linear equation in one variable has no solution.
Estimated Time:2m 0s
Question 52Question →

If 35(2x−4)−12(x−3)=65\frac{3}{5}(2x - 4) - \frac{1}{2}(x - 3) = \frac{6}{5}, what is the value of 5x−45x - 4?

Show answer & explanation

Answer: 11

Answer

11
Evaluating the linear equation by clearing the fractions with a common denominator of 10 gives 6(2x−4)−5(x−3)=126(2x - 4) - 5(x - 3) = 12. Distributing terms yields 12x−24−5x+15=1212x - 24 - 5x + 15 = 12. Combining like terms results in 7x−9=127x - 9 = 12. Solving for xx gives x=3x = 3. Finally, substituting x=3x = 3 into the target expression 5x−45x - 4 yields 5(3)−4=115(3) - 4 = 11.

Step-by-Step Solution

1
Multiply both sides of the equation by 10 to clear the denominators.
6(2x−4)−5(x−3)=126(2x - 4) - 5(x - 3) = 12
The least common denominator of 5 and 2 is 10. Multiplying both sides by 10 simplifies the equation by removing the fractions.
2
Distribute the constants 6 and -5 into the parentheses.
12x−24−5x+15=1212x - 24 - 5x + 15 = 12
Applying the distributive property removes the parentheses. Be careful to distribute the negative sign for the second term: −5⋅(x−3)=−5x+15-5 \cdot (x - 3) = -5x + 15.
3
Combine like terms on the left side of the equation.
7x−9=127x - 9 = 12
Grouping the xx terms (12x−5x=7x12x - 5x = 7x) and the constant terms (−24+15=−9-24 + 15 = -9) simplifies the equation.
4
Isolate the variable term 7x7x and solve for xx.
x=3x = 3
Adding 9 to both sides gives 7x=217x = 21. Dividing both sides by 7 yields x=3x = 3.
5
Substitute the value of xx into the expression 5x−45x - 4.
11
The question asks for the value of 5x−45x - 4, not just xx, so we evaluate 5(3)−4=115(3) - 4 = 11.

Key Concept

Solving linear equations in one variable by clearing fractions, distributing terms, and isolating the variable.
Question 53Question →

A local logistics company charges a flat account setup fee of 55 dollars plus 44 dollars per pound for the first 66 pounds of a package shipment. For any weight beyond 66 pounds, the company charges 33 dollars per pound. If the total cost to ship a package is 5050 dollars, what is the total weight of the package, in pounds?

Show answer & explanation

Answer: 13

Answer

The correct answer is 13, which represents the total weight of the package in pounds.
To find the total weight, we first determine the cost of the first 66 pounds, which is the 55 dollars flat fee plus 44 dollars per pound, totaling 2929 dollars. Subtracting this baseline cost from the total cost of 5050 dollars leaves 2121 dollars for the weight exceeding 66 pounds. Since the rate for the excess weight is 33 dollars per pound, the excess weight is 2121 divided by 33, which is 77 pounds. Adding the initial 66 pounds gives a total weight of 1313 pounds.

Step-by-Step Solution

1
Calculate the total cost of shipping the first 6 pounds of the package.
The cost for the first 66 pounds, including the flat setup fee, is 5+6×4=295 + 6 \times 4 = 29 dollars.
This establishes the baseline cost before the rate of 33 dollars per pound is applied to any weight exceeding 66 pounds.
2
Set up a linear equation representing the total shipping cost of 50 dollars for a package of weight w, where w is greater than 6.
The equation is 29+3(w−6)=5029 + 3(w - 6) = 50.
The total cost is the sum of the cost for the first 66 pounds (2929 dollars) and the cost of the excess weight (w−6w - 6) at the rate of 33 dollars per pound.
3
Solve the linear equation for w by isolating the variable.
3(w−6)=21  ⟹  w−6=7  ⟹  w=133(w - 6) = 21 \implies w - 6 = 7 \implies w = 13.
Subtracting 2929 from both sides and then dividing by 33 isolates the term representing the excess weight, and adding 66 yields the total weight.

Key Concept

Linear Equations in One Variable
Question 54Question →

An online streaming service offers the two monthly subscription plans described in the table below:

PlanMonthly feeCost per premium movie rental
Plan A$12$1.50
Plan B$27 (includes first 4 rentals)$0.75 (for each rental after the first 4)

If a user rented mm premium movies in a month, where m>4m > 4, and the total cost for both plans would be the same, what is the value of mm?

Show answer & explanation

Answer: 16

Answer

The value of mm that results in the same total monthly cost for both plans is 16.
The correct answer of 16 represents the exact number of premium movie rentals where the total monthly cost for both plans is equal to $36. Any other number of movie rentals will result in different costs for the two plans.

Step-by-Step Solution

1
Define the cost equation for Plan A
CostA=12+1.50m\text{Cost}_A = 12 + 1.50m
Plan A charges a flat 12monthlyfeeplus12 monthly fee plus 1.50 for each of the mm movies rented.
2
Define the cost equation for Plan B
CostB=27+0.75(m−4)\text{Cost}_B = 27 + 0.75(m - 4)
Plan B charges a 27monthlyfeethatcoversthefirst4movies,and27 monthly fee that covers the first 4 movies, and 0.75 for each of the m−4m - 4 additional movies rented because m>4m > 4.
3
Equate the two cost expressions and solve for mm
12+1.50m=27+0.75(m−4)  ⟹  12+1.50m=24+0.75m  ⟹  0.75m=12  ⟹  m=1612 + 1.50m = 27 + 0.75(m - 4) \implies 12 + 1.50m = 24 + 0.75m \implies 0.75m = 12 \implies m = 16
To find when the costs are identical, set the two algebraic expressions equal to each other and isolate the variable mm.

Key Concept

Setting up and solving linear equations in one variable from context

Alternative Method

Instead of setting up full equations, we can look at the cost difference at m=4m = 4. At 44 movies, Plan A costs 12+1.50(4)=1812 + 1.50(4) = 18 dollars, and Plan B costs 2727 dollars (since 44 movies are included). The price difference is 27−18=927 - 18 = 9 dollars. For each movie rented beyond 44, the cost of Plan A increases by 1.501.50 dollars while Plan B only increases by 0.750.75 dollars. The rate of change difference is 1.50−0.75=0.751.50 - 0.75 = 0.75 dollars per movie. To bridge the initial 99 dollar difference, the user needs to rent 90.75=12\frac{9}{0.75} = 12 more movies. Thus, the total number of movies is 4+12=164 + 12 = 16.
Estimated Time:1m 30s
Question 55Question →

If 3(2y−5)−4(y−2)=113(2y - 5) - 4(y - 2) = 11, what is the value of 3y+23y + 2?

Show answer & explanation

Answer: 29

Answer

29
The correct value is obtained by first distributing the terms, simplifying the equation to find the value of yy, and then evaluating the expression 3y+23y + 2. Specifically, distributing yields 6y−15−4y+8=116y - 15 - 4y + 8 = 11, which simplifies to 2y−7=112y - 7 = 11. Adding 77 to both sides gives 2y=182y = 18, so y=9y = 9. Substituting y=9y = 9 into 3y+23y + 2 gives 3(9)+2=293(9) + 2 = 29.

Step-by-Step Solution

1
Distribute the coefficients outside the parentheses.
6y−15−4y+8=116y - 15 - 4y + 8 = 11
To simplify the linear equation, we must first remove the parentheses by multiplying the terms inside by the factors outside.
2
Combine like terms on the left side of the equation.
2y−7=112y - 7 = 11
Combining the variable terms (6y−4y=2y6y - 4y = 2y) and constant terms (−15+8=−7-15 + 8 = -7) simplifies the equation.
3
Isolate the variable term by adding 7 to both sides, then divide by 2.
y=9y = 9
Adding 7 gives 2y=182y = 18, and dividing by 2 isolates yy.
4
Substitute the value of yy into the expression 3y+23y + 2.
3(9)+2=293(9) + 2 = 29
The question asks for the value of the expression 3y+23y + 2, not just the variable yy.

Key Concept

Linear Equations in One Variable
Question 56Question →

A drone's battery charge is at 98%98\% when it begins a mission. During the mission, the battery charge decreases by 1.5%1.5\% per minute when the drone is hovering, and by 2.5%2.5\% per minute when it is flying horizontally. The drone flies horizontally for exactly twice as many minutes as it hovers. If the battery charge is at 33%33\% at the end of the mission, for how many minutes did the drone hover?

Show answer & explanation

Answer: 10

Answer

The drone hovered for 10 minutes.
Let tt represent the number of minutes the drone hovers. Since the drone flies horizontally for twice as long as it hovers, it flies horizontally for 2t2t minutes. The total percentage drop in battery is 98−33=6598 - 33 = 65. The equation representing the total decrease in battery charge is 1.5t+2.5(2t)=651.5t + 2.5(2t) = 65. Simplifying the equation yields 1.5t+5t=651.5t + 5t = 65, which becomes 6.5t=656.5t = 65. Dividing both sides of the equation by 6.56.5 gives t=10t = 10.

Step-by-Step Solution

1
Define the variable for the unknown quantity.
Let tt be the number of minutes the drone spent hovering. The time spent flying horizontally is then 2t2t minutes.
The problem states that the horizontal flight time is exactly twice the hovering time.
2
Set up an equation representing the total decrease in battery charge.
The total percentage decrease is 1.5t+2.5(2t)=98−331.5t + 2.5(2t) = 98 - 33.
The battery decreases by 1.5%1.5\% per minute of hovering, 2.5%2.5\% per minute of horizontal flight, and the total change is from 98%98\% to 33%33\%.
3
Simplify the equation and solve for tt.
6.5t=656.5t = 65, which gives t=10t = 10.
Combine like terms and divide both sides by 6.56.5 to isolate the variable.

Key Concept

Setting up and solving a linear equation in one variable from a real-world scenario.
Question 57Question →

If 14(8x−12)−23(3−6x)=13\frac{1}{4}(8x - 12) - \frac{2}{3}(3 - 6x) = 13, what is the value of 3x−23x - 2?

Show answer & explanation

Answer: 7

Answer

7
Distributing the coefficients yields 14(8x−12)=2x−3\frac{1}{4}(8x - 12) = 2x - 3 and −23(3−6x)=−2+4x-\frac{2}{3}(3 - 6x) = -2 + 4x. Combining these terms gives the simplified equation 6x−5=136x - 5 = 13. Adding 55 to both sides yields 6x=186x = 18, and dividing by 66 gives x=3x = 3. Substituting x=3x = 3 into the expression 3x−23x - 2 results in 3(3)−2=73(3) - 2 = 7.

Step-by-Step Solution

1
Distribute the fraction coefficients to the terms inside the parentheses.
14(8x−12)=2x−3\frac{1}{4}(8x - 12) = 2x - 3 and −23(3−6x)=−2+4x-\frac{2}{3}(3 - 6x) = -2 + 4x.
This simplifies the equation by removing the parentheses.
2
Combine like terms on the left side of the equation.
(2x−3)+(−2+4x)=6x−5(2x - 3) + (-2 + 4x) = 6x - 5, transforming the equation to 6x−5=136x - 5 = 13.
Grouping the variable terms and constant terms is necessary to isolate the variable.
3
Isolate the variable by adding 5 to both sides and dividing by 6.
6x=18⇒x=36x = 18 \Rightarrow x = 3.
This determines the value of the variable.
4
Substitute the value of xx into the expression 3x−23x - 2.
3(3)−2=73(3) - 2 = 7.
The question asks for the value of the expression 3x−23x - 2, not the value of xx itself.

Key Concept

Linear Equations in One Variable

Alternative Method

An alternative method is to multiply both sides of the equation by the least common multiple of the denominators 44 and 33, which is 1212. Multiplying the entire equation by 1212 yields 3(8x−12)−8(3−6x)=1563(8x - 12) - 8(3 - 6x) = 156. Distributing gives 24x−36−24+48x=15624x - 36 - 24 + 48x = 156, which simplifies to 72x−60=15672x - 60 = 156. Adding 6060 gives 72x=21672x = 216, so x=3x = 3. Substituting x=3x = 3 into 3x−23x - 2 gives 77.
Estimated Time:1m 30s
Question 58Question →

If 23(6x−9)−34(4x−8)=12(x+10)\frac{2}{3}(6x - 9) - \frac{3}{4}(4x - 8) = \frac{1}{2}(x + 10), what is the value of xx?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10.
Distributing the fractions on the left-hand side of the equation simplifies the expression to xx. Setting this equal to the right-hand side yields the equation x=12(x+10)x = \frac{1}{2}(x + 10). Multiplying both sides by 2 gives 2x=x+102x = x + 10, and subtracting xx from both sides gives the correct value of 10.

Step-by-Step Solution

1
Distribute the fraction 23\frac{2}{3} to the terms inside the first parentheses, (6x−9)(6x - 9), and distribute the fraction −34-\frac{3}{4} to the terms inside the second parentheses, (4x−8)(4x - 8).
4x−6−3x+6=12(x+10)4x - 6 - 3x + 6 = \frac{1}{2}(x + 10)
Applying the distributive property removes the parentheses so that like terms can be combined.
2
Combine the variable terms and constant terms on the left side of the equation.
x=12(x+10)x = \frac{1}{2}(x + 10)
Combining 4x4x and −3x-3x yields xx, while −6-6 and 66 cancel each other out.
3
Multiply both sides of the equation by 2 to eliminate the fraction.
2x=x+102x = x + 10
Multiplying by the denominator simplifies the equation by removing the fraction.
4
Subtract xx from both sides of the equation to isolate the variable.
x=10x = 10
Subtracting xx isolates the variable xx on the left side of the equation.

Key Concept

Solving linear equations in one variable by applying the distributive property and combining like terms.
Question 59Question →

If 5(2a−3)−3(4a−1)=−(a−8)5(2a - 3) - 3(4a - 1) = -(a - 8), what is the value of aa?

Show answer & explanation

Answer: -20

Answer

-20
Distributing the constants on both sides of the equation 5(2a−3)−3(4a−1)=−(a−8)5(2a - 3) - 3(4a - 1) = -(a - 8) yields 10a−15−12a+3=−a+810a - 15 - 12a + 3 = -a + 8. Combining like terms on the left side gives −2a−12=−a+8-2a - 12 = -a + 8. Adding 2a2a to both sides results in −12=a+8-12 = a + 8. Subtracting 88 from both sides gives the correct value of aa, which is −20-20.

Step-by-Step Solution

1
Distribute the coefficients to the terms inside the parentheses on both sides of the equation.
10a−15−12a+3=−a+810a - 15 - 12a + 3 = -a + 8
To eliminate parentheses and allow terms to be combined, apply the distributive property: 5(2a−3)=10a−155(2a - 3) = 10a - 15, −3(4a−1)=−12a+3-3(4a - 1) = -12a + 3, and −(a−8)=−a+8-(a - 8) = -a + 8.
2
Combine like terms on the left side of the equation.
−2a−12=−a+8-2a - 12 = -a + 8
Simplify the left side by combining the variable terms (10a−12a=−2a10a - 12a = -2a) and the constant terms (−15+3=−12-15 + 3 = -12).
3
Isolate the variable aa on one side of the equation.
a=−20a = -20
Add 2a2a to both sides of the equation to get −12=a+8-12 = a + 8, then subtract 88 from both sides to find that a=−20a = -20.

Key Concept

Solving linear equations in one variable using the distributive property, combining like terms, and isolating the variable.
Estimated Time:1m 30s
Question 60Question →

An empty water tank is being filled by two hoses. Hose A fills the tank at a constant rate of 12 gallons per minute. Hose B fills the tank at a constant rate of 15 gallons per minute. Hose A is turned on first, and 5 minutes later, Hose B is also turned on. If both hoses run until the tank contains a total of 330 gallons of water, for how many minutes was Hose A running?

Show answer & explanation

Answer: 15

Answer

15
Let mm be the number of minutes Hose A was running. Since Hose B was turned on 5 minutes after Hose A, Hose B was running for m−5m - 5 minutes. The volume of water contributed by Hose A is 12m12m gallons, and the volume of water contributed by Hose B is 15(m−5)15(m - 5) gallons. The sum of these volumes is 330 gallons, which gives the equation 12m+15(m−5)=33012m + 15(m - 5) = 330. Distributing 15 gives 12m+15m−75=33012m + 15m - 75 = 330. Combining like terms yields 27m−75=33027m - 75 = 330. Adding 75 to both sides gives 27m=40527m = 405. Dividing both sides by 27 yields m=15m = 15. Thus, Hose A was running for 15 minutes.

Step-by-Step Solution

1
Define variables for the running time of each hose.
Let Hose A run for mm minutes. Then Hose B runs for m−5m - 5 minutes.
Hose B started 5 minutes after Hose A, so it ran for 5 fewer minutes.
2
Set up a linear equation for the total gallons of water.
12m+15(m−5)=33012m + 15(m - 5) = 330
The sum of the water from both hoses must equal 330 gallons.
3
Simplify and solve the linear equation.
27m=40527m = 405, which gives m=15m = 15.
Distribute, combine like terms, and isolate the variable mm.

Key Concept

Setting up and solving a multi-step linear equation in one variable from a real-world context.
PreviousPage 3 / 4Next
Linear Equations in One Variable Practice Questions — SAT — Page 3 | Examkin