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Solving One-Variable Linear Equations

53 questions

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Question 41Question →

What is the value of kk that makes the following equation true?

4(k−2)−3k=5k−2(k+10) 4(k - 2) - 3k = 5k - 2(k + 10)
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Answer: 6

Answer

The correct value for the variable is 6.
By following the correct order of operations, distributing carefully, and balancing the equation, we find that k = 6 makes both sides of the equation equal.

Step-by-Step Solution

1
Distribute the values outside the parentheses on both sides of the equation.
4k−8−3k=5k−2k−20 4k - 8 - 3k = 5k - 2k - 20
Applying the distributive property removes the parentheses and allows for combining like terms.
2
Combine like terms on each side of the equation.
k−8=3k−20 k - 8 = 3k - 20
Simplifying each side individually makes it easier to isolate the variable.
3
Subtract kk from both sides to collect the variables on the right side.
−8=2k−20 -8 = 2k - 20
Balancing the equation by performing the inverse operation isolates the variable terms on one side.
4
Add 20 to both sides to isolate the variable term.
12=2k 12 = 2k
Moving the constants to the opposite side of the variable completes the isolation process.
5
Divide both sides by 2.
k=6 k = 6
Dividing by the coefficient gives the final value of the variable.

Key Concept

Solving multi-step linear equations involves distributing, combining like terms, and balancing the equation using inverse operations.
Question 42Question →

An algebraic equation requires simplification to isolate the unknown variable. Which of the following represents the correct value of yy in the equation below?

3(2y+5)−y=2(2y−1)+13 3(2y + 5) - y = 2(2y - 1) + 13
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Answer: -4

Answer

The correct value for yy is −4-4.
The correct answer is found by properly applying the distributive property to both sides, combining the resulting like terms, and systematically isolating the variable. Maintaining the balance of the equation by performing inverse operations yields a final value of -4.

Step-by-Step Solution

1
Distribute the constants into the parentheses on both sides of the equation.
6y+15−y=4y−2+136y + 15 - y = 4y - 2 + 13
To eliminate the parentheses and prepare the equation for combining terms.
2
Combine like terms on each individual side.
5y+15=4y+115y + 15 = 4y + 11
To simplify the expressions on both sides before isolating the variable.
3
Subtract 4y4y from both sides.
y+15=11y + 15 = 11
To group all variable terms onto the left side of the equation.
4
Subtract 1515 from both sides.
y=−4y = -4
To completely isolate yy and find the final root of the equation.

Key Concept

Solving one-variable linear equations by applying the distributive property and combining like terms.
Question 43Question →

What value of xx satisfies the equation 12(4x−6)+3x=23(9x−12)\frac{1}{2}(4x - 6) + 3x = \frac{2}{3}(9x - 12)?

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Answer: 5

Answer

5
The correct answer is 55. By distributing the fractions, combining the like terms, and properly using inverse operations to isolate xx, the equation simplifies step-by-step to x=5x = 5.

Step-by-Step Solution

1
Distribute the fractions into the parentheses on both sides of the equation.
2x−3+3x=6x−82x - 3 + 3x = 6x - 8
Applying the distributive property removes the parentheses and prepares the equation for combining terms.
2
Combine the like terms on the left side.
5x−3=6x−85x - 3 = 6x - 8
Simplifying each side individually makes it easier to isolate the variable.
3
Subtract 5x5x from both sides to group the variable terms on the right side.
−3=x−8-3 = x - 8
Keeping the variable coefficient positive reduces the chance of a sign error.
4
Add 88 to both sides to isolate xx.
5=x5 = x
Applying the inverse operation isolates the variable and provides the final solution.

Key Concept

Solving One-Variable Linear Equations
Question 44Question →

Solve the following linear equation for ww:

13(12w−9)+5=2(w−1)+8 \frac{1}{3}(12w - 9) + 5 = 2(w - 1) + 8

Enter the exact integer value that satisfies the equation.

Fill in the blanks below

The value of ww is .
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Answer

2
By successfully distributing the factors, combining like terms on each side, and balancing the equation by moving variables and constants to opposite sides, the isolated variable equates to exactly 2.

Step-by-Step Solution

1
Distribute the coefficients to remove parentheses on both sides of the equation.
4w−3+5=2w−2+84w - 3 + 5 = 2w - 2 + 8
Applying the distributive property simplifies the expression and exposes the variable terms.
2
Combine the constant terms on each side.
4w+2=2w+64w + 2 = 2w + 6
Simplifying each side makes it easier to balance the equation in the next steps.
3
Subtract 2w2w from both sides of the equation.
2w+2=62w + 2 = 6
This groups all the variable terms onto the left side.
4
Subtract 22 from both sides, then divide the result by 22.
2w=4⇒w=22w = 4 \Rightarrow w = 2
This completely isolates the variable ww to reveal its final value.

Key Concept

Solving One-Variable Linear Equations
Question 45Question →

Evaluate the given algebraic equation to solve for pp:

7−3(p+2)=p−77 - 3(p + 2) = p - 7

Which of the following represents the correct solution for pp?

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Answer: 2

Answer

The correct value is 2.
The correct approach involves distributing the −3-3 into the parentheses to get 7−3p−67 - 3p - 6, combining like terms to get 1−3p1 - 3p, and then balancing the equation to solve for pp. Adding 3p3p to both sides yields 1=4p−71 = 4p - 7, and adding 77 gives 8=4p8 = 4p, which simplifies to p=2p = 2.

Step-by-Step Solution

1
Apply the distributive property to the left side of the equation.
7−3p−6=p−77 - 3p - 6 = p - 7
The -3 must be multiplied by both pp and 22 to eliminate the parentheses.
2
Combine the constant terms on the left side.
1−3p=p−71 - 3p = p - 7
Combining 77 and −6-6 simplifies the equation before moving terms across the equals sign.
3
Add 3p3p to both sides of the equation.
1=4p−71 = 4p - 7
This groups all terms containing the variable pp on the right side.
4
Add 77 to both sides and divide by 44.
8=4p⇒p=28 = 4p \Rightarrow p = 2
This isolates the variable pp to find the final numerical solution.

Key Concept

Solving Linear Equations with Distribution
Question 46Question →

Which of the following represents the value of yy that makes the given equation true?

2y−53+8=3\frac{2y - 5}{3} + 8 = 3
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Answer: -5

Answer

The correct value for the variable is -5.
The solution requires a sequence of inverse operations to isolate the variable. First, subtracting 8 from both sides gives a value of -5 on the right. Then, multiplying by 3 clears the fraction, resulting in 2y - 5 = -15. Adding 5 yields 2y = -10, and dividing by 2 confirms the final value is -5.

Step-by-Step Solution

1
Isolate the fractional term by subtracting 8 from both sides of the equation.
2y−53=−5\frac{2y - 5}{3} = -5
Applying the inverse operation of addition moves the constant term to the right side, simplifying the equation.
2
Multiply both sides of the equation by 3.
2y−5=−152y - 5 = -15
Multiplying by the denominator eliminates the fraction, making it possible to solve for the variable.
3
Add 5 to both sides of the equation.
2y=−102y = -10
This step isolates the term with the variable on the left side.
4
Divide both sides by 2.
y=−5y = -5
Dividing by the coefficient of the variable gives the final exact value.

Key Concept

Solving one-variable linear equations involving fractions
Question 47Question →

Consider the following mathematical relationship:

34(m−8)+2=12m−1 \frac{3}{4}(m - 8) + 2 = \frac{1}{2}m - 1

Which of the following represents the correct solution for mm?

Show answer & explanation

Answer: 12

Answer

The correct solution is 12.
The correct solution is found by systematically applying the distributive property to eliminate parentheses, combining the constant terms, grouping the variable terms on one side, and fully isolating the variable. Following these steps accurately yields a value of 12.

Step-by-Step Solution

1
Apply the distributive property to the left side of the equation.
34m−6+2=12m−1 \frac{3}{4}m - 6 + 2 = \frac{1}{2}m - 1
This removes the parentheses by multiplying the fraction through the terms inside.
2
Combine the constant terms on the left side.
34m−4=12m−1 \frac{3}{4}m - 4 = \frac{1}{2}m - 1
Simplifying the expressions on both sides makes it easier to isolate the variable.
3
Subtract 12m \frac{1}{2}m from both sides to group the variable terms.
14m−4=−1 \frac{1}{4}m - 4 = -1
This places all terms containing the variable on one side of the equation. (Note: 34m−24m=14m \frac{3}{4}m - \frac{2}{4}m = \frac{1}{4}m )
4
Add 4 to both sides to isolate the variable term.
14m=3 \frac{1}{4}m = 3
This cancels out the constant on the left side, leaving only the variable term.
5
Multiply both sides by 4.
m=12 m = 12
This clears the fractional coefficient and solves for the final value of the variable.

Key Concept

Solving one-variable linear equations with fractions and distribution
Question 48Question →

A fitness center offers two membership plans. Plan A requires a one-time enrollment fee of 4545 dollars and costs 3030 dollars per month. Plan B requires a one-time enrollment fee of 105105 dollars and costs 2020 dollars per month. For how many months of membership will the total cost of both plans be exactly the same?

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Answer: 6

Answer

The plans will cost the exact same amount at 6 months.
Setting the cost of Plan A (45+30m45 + 30m) equal to the cost of Plan B (105+20m105 + 20m) creates the equation 45+30m=105+20m45 + 30m = 105 + 20m. By isolating the variable, we find that 10m=6010m = 60, which simplifies to m=6m = 6. Therefore, it takes exactly 66 months for both memberships to cost the same.

Step-by-Step Solution

1
Define the variable and formulate a one-variable linear equation.
45+30m=105+20m45 + 30m = 105 + 20m
To determine when the total costs are equal, the cost expression for Plan A (4545 fee plus 3030 per month) must be set equal to the cost expression for Plan B (105105 fee plus 2020 per month).
2
Subtract 20m20m from both sides of the equation.
45+10m=10545 + 10m = 105
This groups all terms containing the variable mm on one side of the equation.
3
Subtract 4545 from both sides of the equation.
10m=6010m = 60
This isolates the variable term by moving all constant values to the opposite side of the equation.
4
Divide both sides by 1010.
m=6m = 6
Dividing by the coefficient of the variable provides the final numerical solution for the unknown.

Key Concept

Translating a real-world scenario into a one-variable linear equation and isolating the variable to solve for an unknown value.
Estimated Time:1m 30s
Question 49Question →

Which of the following values for mm makes the equation true?

35(10m−15)−4m=12(8m+14)−34 \frac{3}{5}(10m - 15) - 4m = \frac{1}{2}(8m + 14) - 34
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Answer: 9

Answer

The correct answer is 9.
The correct answer is found by properly executing the distributive property on both sides to clear the parentheses. On the left side, 3/5 is distributed to yield 6m - 9. On the right side, 1/2 is distributed to yield 4m + 7. After combining the remaining like terms, the equation simplifies to 2m - 9 = 4m - 27. Adding 27 and subtracting 2m from both sides gives 18 = 2m, which simplifies to m = 9.

Step-by-Step Solution

1
Distribute the fractions into the parentheses on both sides of the equation.
6m−9−4m=4m+7−346m - 9 - 4m = 4m + 7 - 34
The distributive property must be applied first to expand the expressions and eliminate the parentheses.
2
Combine like terms on each side of the equal sign.
2m−9=4m−272m - 9 = 4m - 27
Simplifying each side individually makes it easier to isolate the variable in subsequent steps.
3
Subtract 2m from both sides.
−9=2m−27-9 = 2m - 27
This gathers all variable terms on one side of the equation.
4
Add 27 to both sides and divide the result by 2.
18=2m18 = 2m, which means m=9m = 9
This isolates the variable completely and provides the final solution.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:1m 30s
Question 50Question →

Solve the following equation for the variable nn:

2n−43=n+52\frac{2n - 4}{3} = \frac{n + 5}{2}

What is the correct value of nn?

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Answer: 23

Answer

The correct value for the variable is 23.
The correct value is 23, which is found by properly cross-multiplying the fractions, distributing the values to all terms in the numerators, and maintaining equality while isolating the variable.

Step-by-Step Solution

1
Cross-multiply to eliminate the fractions.
2(2n−4)=3(n+5)2(2n - 4) = 3(n + 5)
Multiplying each side's numerator by the other side's denominator clears the fractions from the equation.
2
Distribute the constants on both sides of the equation.
4n−8=3n+154n - 8 = 3n + 15
Applying the distributive property removes the parentheses so like terms can be combined.
3
Subtract 3n3n from both sides.
n−8=15n - 8 = 15
This moves all variable terms to the left side of the equation to begin isolating nn.
4
Add 8 to both sides.
n=23n = 23
This isolates the variable nn to find the final solution.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:1m 30s
Question 51Question →

If −4(2n−3)=2n−3(3n+1)-4(2n - 3) = 2n - 3(3n + 1), what is the value of nn?

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Answer: 15

Answer

15
The equation correctly simplifies to -8n + 12 = -7n - 3 after proper distribution and combining of like terms. Isolating the variable by adding 8n and 3 to both sides yields n = 15.

Step-by-Step Solution

1
Distribute the factors on both sides of the equation.
−8n+12=2n−9n−3-8n + 12 = 2n - 9n - 3
Multiplying the outside numbers across the terms inside the parentheses removes the grouping symbols.
2
Combine like terms on the right side.
−8n+12=−7n−3-8n + 12 = -7n - 3
Adding 2n2n and −9n-9n simplifies the right side to a single variable term.
3
Isolate the variable nn.
12+3=−7n+8n12 + 3 = -7n + 8n, which simplifies to 15=n15 = n
Moving all variable terms to one side and constant terms to the other side solves the equation.

Key Concept

Solving one-variable linear equations requiring distribution and combining like terms.
Question 52Question →

Find the value of xx that satisfies the equation:

x−34+2x+13=5x6 \frac{x - 3}{4} + \frac{2x + 1}{3} = \frac{5x}{6}

Enter the exact value of xx.

Show answer & explanation

Answer: 5

Answer

5
By finding the least common denominator (12) and multiplying every term by it, the fractions are cleanly eliminated. After applying the distributive property and combining like terms (11x−5=10x11x - 5 = 10x), isolating the variable xx yields exactly 5.

Step-by-Step Solution

1
Multiply all terms by the least common denominator (12).
12(x−34)+12(2x+13)=12(5x6)12 \left(\frac{x - 3}{4}\right) + 12 \left(\frac{2x + 1}{3}\right) = 12 \left(\frac{5x}{6}\right)
Multiplying by the LCD eliminates all fractions from the equation, simplifying further steps.
2
Simplify the resulting fractions.
3(x−3)+4(2x+1)=2(5x)3(x - 3) + 4(2x + 1) = 2(5x)
Dividing 12 by each respective denominator leaves an integer multiplier for the numerators.
3
Distribute the multipliers across the terms in the parentheses.
3x−9+8x+4=10x3x - 9 + 8x + 4 = 10x
The distributive property is required to remove the parentheses.
4
Combine like terms on the left side of the equation.
11x−5=10x11x - 5 = 10x
Grouping xx terms (3x+8x3x + 8x) and constant terms (−9+4-9 + 4) simplifies the expression.
5
Isolate the variable xx.
x=5x = 5
Subtracting 10x10x from both sides gives x−5=0x - 5 = 0. Adding 55 to both sides results in x=5x = 5.

Key Concept

Solving linear equations involving rational expressions by multiplying by a least common denominator.
Estimated Time:1m 30s
Question 53Question →

Calculate the value of ww in the given linear equation:

Fill in the blanks below

3(2w+5)−4w=5(w−1)+2 3(2w + 5) - 4w = 5(w - 1) + 2

Value of ww:
Show answer & explanation

Answer

The correct value for the variable is 6.
The correct answer is 6. Applying the distributive property to both sides simplifies the equation to 6w+15−4w=5w−5+26w + 15 - 4w = 5w - 5 + 2. Grouping like terms on each side yields 2w+15=5w−32w + 15 = 5w - 3. Subtracting 2w2w from both sides and adding 33 to both sides results in 18=3w18 = 3w. Dividing both sides by 3 isolates the variable, giving w=6w = 6.

Step-by-Step Solution

1
Distribute the constants across the terms inside the parentheses on both sides.
6w+15−4w=5w−5+26w + 15 - 4w = 5w - 5 + 2
To eliminate the parentheses and expose all terms for simplification.
2
Combine the like terms on each side of the equation independently.
2w+15=5w−32w + 15 = 5w - 3
To simplify both expressions before manipulating terms across the equals sign.
3
Move all variable terms to one side of the equation.
Subtract 2w2w from both sides to obtain 15=3w−315 = 3w - 3.
To group all the terms containing ww together and avoid negative coefficients.
4
Isolate the variable term and solve for ww.
Add 3 to both sides to get 18=3w18 = 3w, then divide by 3 to find w=6w = 6.
To determine the final numerical value of the variable.

Key Concept

Solving multi-step linear equations utilizing the distributive property and combining like terms.
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