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Solving One-Variable Linear Equations

53 questions

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Question 21Question →

If 4(k−3)=284(k - 3) = 28, what is the value of kk?

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Answer: 10

Answer

10
Applying inverse operations to isolate the variable results in k=10k = 10. This can be verified by substituting 10 back into the original equation: 4(10−3)=4(7)=284(10 - 3) = 4(7) = 28, which makes the statement true.

Step-by-Step Solution

1
Divide both sides by 4.
k−3=7k - 3 = 7
Dividing by the coefficient outside the parentheses simplifies the equation and isolates the binomial.
2
Add 3 to both sides of the equation.
k=10k = 10
Using the inverse operation of subtraction isolates the variable kk.

Key Concept

Solving One-Variable Linear Equations
Question 22Question →

Determine the value of yy that makes the following equation true:

6y−12=36 6y - 12 = 36

Which of the following represents the correct value of yy?

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Answer: 8

Answer

The correct value of y is 8.
The correct value is 8. To isolate the variable y, the constant 12 must first be added to both sides of the equation, which results in 6y = 48. Finally, dividing both sides by the coefficient 6 gives the solution y = 8.

Step-by-Step Solution

1
Add 12 to both sides of the equation.
6y=36+12⇒6y=48 6y = 36 + 12 \Rightarrow 6y = 48
This isolates the term with the variable on one side of the equation.
2
Divide both sides by 6.
y=486⇒y=8 y = \frac{48}{6} \Rightarrow y = 8
This isolates the variable y and provides the final solution.

Key Concept

Solving two-step linear equations by performing inverse operations.
Question 23Question →

Determine the value of the unknown variable in the following linear equation.

Fill in the blanks below

Solve the equation for mm:
4m−7=13 4m - 7 = 13

The correct value for mm is .
Show answer & explanation

Answer

The value of the variable m is 5.
Applying the inverse operation by adding 7 to both sides gives 4m=204m = 20. Dividing by 4 isolates the variable, resulting in the correct value of 5.

Step-by-Step Solution

1
Add 7 to both sides of the equation.
4m=13+7⇒4m=204m = 13 + 7 \Rightarrow 4m = 20
To isolate the term with the variable (4m4m) on one side of the equation.
2
Divide both sides by 4.
m=204⇒m=5m = \frac{20}{4} \Rightarrow m = 5
To solve for mm by canceling out its coefficient.

Key Concept

Solving one-variable linear equations by applying inverse operations to isolate the variable.
Question 24Question →
Calculate the value of xx that satisfies the given equation:
7x−14=21 7x - 14 = 21
Which of the following represents the correct value?
Show answer & explanation

Answer: 5

Answer

5
The correct answer is 5. To solve the equation, the first step is to add 14 to both sides, which balances the equation and gives 7x = 35. The final step is dividing both sides by 7 to isolate the variable, resulting in x = 5.

Step-by-Step Solution

1
Add 14 to both sides of the equation.
7x=21+14 7x = 21 + 14
7x=35 7x = 35
This isolates the term with the variable by removing the constant on the left side.
2
Divide both sides by 7.
x=5 x = 5
This solves for x by reversing the multiplication by 7.

Key Concept

Solving One-Variable Linear Equations requires isolating the variable using inverse operations.
Question 25Question →

Solve for nn in the equation below:

8n−11=37 8n - 11 = 37

Enter the exact value of nn.

Show answer & explanation

Answer: 6

Answer

6
To find the value of nn, first perform the inverse operation of subtraction by adding 11 to both sides, which simplifies the equation to 8n=488n = 48. Then, perform the inverse operation of multiplication by dividing both sides by 8, yielding n=6n = 6.

Step-by-Step Solution

1
Isolate the variable term.
8n = 48
Adding 11 to both sides neutralizes the -11 on the left side of the equation.
2
Solve for the variable.
n = 6
Dividing both sides by 8 isolates n completely.

Key Concept

Solving two-step linear equations using inverse operations.
Question 26Question →
Find the solution to the following algebraic equation:
4(3r−6)=48 4(3r - 6) = 48
Which of the following is the correct value of rr?
Show answer & explanation

Answer: 6

Answer

The correct value is 6.
The correct answer is found by first distributing the 4 to get 12r - 24 = 48. Next, adding 24 to both sides gives 12r = 72. Finally, dividing both sides by 12 yields r = 6.

Step-by-Step Solution

1
Distribute the 4 into the terms inside the parentheses.
12r−24=48 12r - 24 = 48
To eliminate the parentheses and simplify the left side of the equation.
2
Add 24 to both sides of the equation.
12r=72 12r = 72
To isolate the term containing the variable rr on one side of the equation using inverse operations.
3
Divide both sides of the equation by 12.
r=6 r = 6
To solve for the final, single value of the variable rr.

Key Concept

Solving One-Variable Linear Equations
Question 27Question →

Read the scenario below and solve for the unknown value. Enter your exact numerical answer in the blank provided.

Fill in the blanks below

A mechanic orders a set of spark plugs that cost 8each,plusaflatshippingfeeof8 each, plus a flat shipping fee of 12. If the total charge is $60, the mechanic ordered spark plugs.
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Answer

6
By setting up the equation 8x+12=608x + 12 = 60, we can isolate xx. Subtracting the constant 1212 from both sides gives 8x=488x = 48. Dividing by 88 yields x=6x = 6. Therefore, 66 spark plugs were ordered.

Step-by-Step Solution

1
Set up a linear equation to represent the word problem.
8x+12=608x + 12 = 60
Let xx be the number of spark plugs. Multiplying the cost per plug (8)by8) by x andaddingtheflatshippingfee( and adding the flat shipping fee ( 12) equals the total charge ($60).
2
Subtract 12 from both sides of the equation.
8x=488x = 48
To begin isolating the variable xx, the constant term must be removed from the left side of the equation.
3
Divide both sides by 8.
x=6x = 6
To completely isolate xx, divide by its coefficient. This reveals the number of spark plugs ordered.

Key Concept

Solving one-variable linear equations from a real-world scenario.
Question 28Question →
What value of mm makes the equation 12m−7=41 12m - 7 = 41 true?
Show answer & explanation

Answer: 4

Answer

4
Applying the correct sequence of inverse operations (first adding 7, then dividing by 12) yields exactly 4.

Step-by-Step Solution

1
Add 7 to both sides of the equation.
12m=48 12m = 48
This is the first inverse operation needed to isolate the variable term.
2
Divide both sides by 12.
m=4 m = 4
This inverse operation isolates mm, giving the final solution.

Key Concept

Solving two-step linear equations using inverse operations
Estimated Time:30s
Question 29Question →

Consider the linear equation below:

6y−18=24 6y - 18 = 24

What is the value of yy?

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Answer: 7

Answer

The correct value is 7.
The correct answer is found by first adding 18 to both sides of the equation to get 6y=426y = 42. Then, dividing both sides by 6 isolates yy, resulting in y=7y = 7.

Step-by-Step Solution

1
Isolate the term with the variable by adding 18 to both sides of the equation.
6y=42 6y = 42
Applying the addition property of equality cancels out the -18 on the left side.
2
Solve for the variable by dividing both sides by 6.
y=7 y = 7
Dividing by the coefficient of the variable isolates yy and provides the final solution.

Key Concept

Solving linear equations by isolating the variable using inverse operations.
Question 30Question →

Find the value of tt that satisfies the linear equation below.

Fill in the blanks below

t3+8=15 \frac{t}{3} + 8 = 15

The value of tt is .
Show answer & explanation

Answer

21
To find the value of tt, you must apply inverse operations to isolate the variable. First, subtract 8 from both sides to get t/3=7t/3 = 7. Then, multiply both sides by 3 to eliminate the fraction, which gives the final solution t=21t = 21.

Step-by-Step Solution

1
Isolate the fraction term by subtracting 8 from both sides of the equation.
t3=15−8 \frac{t}{3} = 15 - 8
t3=7 \frac{t}{3} = 7
To solve for a variable, you must reverse the order of operations. Addition is reversed first using subtraction to isolate the term containing the variable.
2
Solve for tt by multiplying both sides of the equation by 3.
t=7×3 t = 7 \times 3
t=21 t = 21
Multiplying by the denominator cancels out the fraction, leaving the variable tt by itself.

Key Concept

Solving two-step linear equations using inverse operations.
Question 31Question →

Solve the linear equation 10−4p=−1810 - 4p = -18. Which of the following is the value of pp?

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Answer: 7

Answer

The correct value for p is 7.
The correct answer is 7 because applying inverse operations accurately isolates the variable. Subtracting 10 from both sides yields −4p=−28-4p = -28, and dividing both sides by −4-4 results in p=7p = 7.

Step-by-Step Solution

1
Subtract 10 from both sides of the equation.
−4p=−18−10-4p = -18 - 10 which simplifies to −4p=−28-4p = -28
This isolates the term containing the variable pp on one side of the equation.
2
Divide both sides by -4.
p=−28−4p = \frac{-28}{-4} which simplifies to p=7p = 7
This removes the coefficient from pp, giving the final solution.

Key Concept

Solving a one-variable linear equation requires isolating the variable by performing inverse operations equally on both sides.
Question 32Question →

An algebraic equation is shown below.

4(x−3)−2x=5x−27 4(x - 3) - 2x = 5x - 27

Which of the following represents the correct value of xx?

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Answer: 5

Answer

5
The correct answer is derived by first distributing the 4 to get 4x−124x - 12. Combining 4x4x and −2x-2x yields 2x−122x - 12. Moving the variables to one side (5x−2x=3x5x - 2x = 3x) and the constants to the other (−12+27=15-12 + 27 = 15) results in 15=3x15 = 3x. Dividing by 3 gives the final result of x=5x = 5.

Step-by-Step Solution

1
Distribute the coefficient 4 into the parentheses.
4x−12−2x=5x−27 4x - 12 - 2x = 5x - 27
According to the distributive property, the number outside the parentheses must multiply each term inside.
2
Combine the like terms on the left side of the equation.
2x−12=5x−27 2x - 12 = 5x - 27
The terms 4x4x and −2x-2x are like terms and subtract to form 2x2x.
3
Isolate the variable terms on one side by subtracting 2x2x from both sides.
−12=3x−27 -12 = 3x - 27
Subtracting 2x2x from both sides keeps the equation balanced and groups all terms containing xx on the right side.
4
Isolate the variable xx by adding 27 to both sides.
15=3x 15 = 3x
Adding 27 cancels out the −27-27 on the right, leaving the variable term by itself.
5
Solve for xx by dividing both sides by 3.
x=5 x = 5
Dividing by the coefficient of xx provides the final value of the unknown variable.

Key Concept

Solving linear equations requires correctly applying the distributive property, combining like terms accurately, and using inverse operations equally on both sides to isolate the variable.
Question 33Question →

Calculate the precise value of kk that makes the following algebraic equation true:

5(2k−3)=3k+13 5(2k - 3) = 3k + 13
Show answer & explanation

Answer: 4

Answer

4
The correct value is found by distributing the 5, gathering all variable terms on one side and constants on the other, and finally dividing by the variable's coefficient.

Step-by-Step Solution

1
Apply the distributive property on the left side of the equation.
10k−15=3k+1310k - 15 = 3k + 13
To eliminate the parentheses and expose the variable terms for grouping.
2
Subtract 3k3k from both sides.
7k−15=137k - 15 = 13
To collect all terms containing the variable kk on one side of the equation.
3
Add 1515 to both sides.
7k=287k = 28
To isolate the variable term by moving constants to the opposite side.
4
Divide both sides by 77.
k=4k = 4
To solve for the final value of kk.

Key Concept

Solving linear equations with variables on both sides using the distributive property.
Question 34Question →

Calculate the unknown value in the given equation:

5n−15=20 5n - 15 = 20

What is the value of nn?

Show answer & explanation

Answer: 7

Answer

7
To solve for n, the constant -15 must be canceled out by adding 15 to both sides, which results in 5n = 35. Then, dividing both sides by the coefficient 5 isolates n, giving the correct value of 7.

Step-by-Step Solution

1
Add 15 to both sides of the equation.
5n−15+15=20+15 5n - 15 + 15 = 20 + 15
5n=35 5n = 35
This isolates the variable term on one side of the equation by using the inverse operation of subtraction.
2
Divide both sides of the equation by 5.
5n5=355 \frac{5n}{5} = \frac{35}{5}
n=7 n = 7
This isolates the variable n completely, yielding the final numerical value.

Key Concept

Solving one-variable linear equations requires using inverse operations to isolate the unknown variable while maintaining balance across the equals sign.
Question 35Question →

Solve the following linear equation for the variable ww:

7−3(2w−4)=w+5 7 - 3(2w - 4) = w + 5

What is the value of ww?

Fill in the blanks below

The value of ww is .
Show answer & explanation

Answer

2
The equation is solved by correctly applying the distributive property to get 7−6w+12=w+57 - 6w + 12 = w + 5, combining like terms to get 19−6w=w+519 - 6w = w + 5, and then using inverse operations to isolate ww, which yields w=2w = 2.

Step-by-Step Solution

1
Distribute the -3 across the terms inside the parentheses.
7−6w+12=w+57 - 6w + 12 = w + 5
The distributive property is required to expand the expression −3(2w−4)-3(2w - 4) before combining like terms. Remember that multiplying a negative by a negative results in a positive.
2
Combine the constant terms on the left side of the equation.
19−6w=w+519 - 6w = w + 5
Adding 77 and 1212 simplifies the left side.
3
Add 6w6w to both sides of the equation.
19=7w+519 = 7w + 5
This step groups all terms containing the variable ww onto the right side.
4
Subtract 5 from both sides.
14=7w14 = 7w
This isolates the variable term 7w7w on one side of the equation.
5
Divide both sides by 7.
w=2w = 2
Dividing by the coefficient of ww solves for the final value.

Key Concept

Solving linear equations involving the distributive property with negative coefficients and variables on both sides.
Estimated Time:1m 0s
Question 36Question →

An equation is given as:

2(3h−5)=4(h+1)−6 2(3h - 5) = 4(h + 1) - 6

What is the value of hh?

Show answer & explanation

Answer: 4

Answer

The correct value of hh is 4.
The correct answer is found by properly distributing the values into both sets of parentheses, combining the constants on the right side correctly (+4 - 6 = -2), and then maintaining the balance of the equation while isolating the variable using inverse operations.

Step-by-Step Solution

1
Distribute the coefficients to the terms inside the parentheses on both sides of the equation.
6h−10=4h+4−66h - 10 = 4h + 4 - 6
To eliminate the parentheses according to the distributive property.
2
Combine the constant terms on the right side of the equation.
6h−10=4h−26h - 10 = 4h - 2
To simplify the right side by calculating +4−6+4 - 6.
3
Subtract 4h4h from both sides of the equation.
2h−10=−22h - 10 = -2
To group all variable terms containing hh onto the left side.
4
Add 10 to both sides of the equation.
2h=82h = 8
To isolate the variable term by canceling out the −10-10 on the left side.
5
Divide both sides by 2.
h=4h = 4
To solve for the final value of hh.

Key Concept

Solving One-Variable Linear Equations
Question 37Question →

Determine the value of the variable yy that satisfies the given linear equation.

Fill in the blanks below

Given the equation 5(y−2)+3y=4y+26 5(y - 2) + 3y = 4y + 26 , the value of yy is .
Show answer & explanation

Answer

9
By distributing the 55 to obtain 5y−105y - 10, combining the yy terms on the left to get 8y8y, and subsequently isolating yy through basic inverse operations (8y−4y=26+108y - 4y = 26 + 10), we determine that 4y=364y = 36, making y=9y = 9.

Step-by-Step Solution

1
Distribute the 5 to the terms inside the parentheses.
5y−10+3y=4y+265y - 10 + 3y = 4y + 26
Applying the distributive property removes the parentheses, allowing like terms to be combined.
2
Combine the like terms on the left side of the equation.
8y−10=4y+268y - 10 = 4y + 26
Simplifying each side individually makes it easier to isolate the variable.
3
Subtract 4y4y from both sides of the equation.
4y−10=264y - 10 = 26
This groups all terms containing the variable yy on one side of the equation.
4
Add 10 to both sides of the equation.
4y=364y = 36
This isolates the variable term by removing the constant from the left side.
5
Divide both sides by 4.
y=9y = 9
Dividing by the coefficient of yy yields the final value.

Key Concept

Solving linear equations with variables on both sides using the distributive property
Question 38Question →

A technician is testing two different electrical circuits. The voltage in Circuit A is given by the expression 5v−125v - 12, where vv is a variable setting on the power supply. The voltage in Circuit B is given by 2(v+9)2(v + 9). Both circuits are adjusted to have the exact same voltage. Fill in the blanks with the correct numerical values.

Fill in the blanks below

The required setting vv is and the resulting voltage is .
Show answer & explanation

Answer

The setting vv is 10 and the resulting voltage is 38.
By setting the two expressions equal, expanding the parentheses, and systematically isolating vv, we find the setting must be 10. Plugging 10 back into the expression evaluates the voltage to 38 for both circuits.

Step-by-Step Solution

1
Set the expressions for Circuit A and Circuit B equal to each other.
5v−12=2(v+9)5v - 12 = 2(v + 9)
The problem states that both circuits are adjusted to have the exact same voltage.
2
Distribute the 2 on the right side of the equation.
5v−12=2v+185v - 12 = 2v + 18
The distributive property is required to eliminate parentheses and expose the variable term.
3
Subtract 2v2v from both sides of the equation.
3v−12=183v - 12 = 18
To consolidate all variable terms on one side of the equation.
4
Add 12 to both sides of the equation.
3v=303v = 30
To isolate the variable term by moving constants to the other side.
5
Divide both sides by 3.
v=10v = 10
To solve for the variable vv.
6
Substitute v=10v = 10 back into either original expression.
5(10)−12=50−12=385(10) - 12 = 50 - 12 = 38
To find the resulting voltage as requested by the second blank.

Key Concept

Solving linear equations with variables on both sides and distributive property
Question 39Question →

Consider the following equation:

4(v−2)+6=7v−2(v+5) 4(v - 2) + 6 = 7v - 2(v + 5)

What is the value of vv that satisfies this equation?

Show answer & explanation

Answer: 8

Answer

8
Expanding the terms by distributing gives 4v - 8 + 6 = 7v - 2v - 10. Simplifying both sides yields 4v - 2 = 5v - 10. By subtracting 4v from both sides and adding 10 to both sides, the variable v is isolated to yield 8.

Step-by-Step Solution

1
Distribute the constants into the parentheses on both sides of the equation.
4v - 8 + 6 = 7v - 2v - 10
This removes the parentheses and expands the expressions so like terms can be combined.
2
Combine the constant terms on the left side and the variable terms on the right side.
4v - 2 = 5v - 10
Simplifying both sides makes it easier to isolate the variable.
3
Subtract 4v from both sides to group the variable terms on the right side.
-2 = v - 10
This step groups the variables together while keeping the coefficient of v positive.
4
Add 10 to both sides to solve for v.
8 = v
This step fully isolates v, giving the final solution.

Key Concept

Solving linear equations involves distributing constants to expand expressions, combining like terms, and performing identical operations on both sides to isolate the unknown variable.
Question 40Question →

Solve for tt in the equation below:

3(2t−4)+5=5t−(t−9) 3(2t - 4) + 5 = 5t - (t - 9)

Fill in the blanks below

The value of tt is .
Show answer & explanation

Answer

The correct value for tt is 8.
By properly distributing values into the parentheses, combining like terms, and isolating the variable using inverse operations, the equation simplifies to 2t=162t = 16, which yields t=8t = 8.

Step-by-Step Solution

1
Distribute the 3 into the parentheses on the left side, and distribute the negative sign into the parentheses on the right side.
6t−12+5=5t−t+96t - 12 + 5 = 5t - t + 9
To eliminate the parentheses so that all terms can be properly combined.
2
Combine the constant terms on the left side and the like variable terms on the right side.
6t−7=4t+96t - 7 = 4t + 9
To simplify both sides of the equation independently before moving terms across the equals sign.
3
Subtract 4t4t from both sides of the equation.
2t−7=92t - 7 = 9
To gather all the variable terms onto the left side.
4
Add 7 to both sides of the equation.
2t=162t = 16
To isolate the term containing the variable.
5
Divide both sides by 2.
t=8t = 8
To find the final numerical value of tt.

Key Concept

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