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Solving One-Variable Linear Equations

53 questions

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Question 1Question →

Consider the linear equation provided below:

−2(y−8)+3y=4(y−1)−y+10 -2(y - 8) + 3y = 4(y - 1) - y + 10

Which of the following numbers correctly satisfies this equation?

Show answer & explanation

Answer: 5

Answer

The correct value that satisfies the equation is 5.
The correct answer is found by first distributing the values outside the parentheses, maintaining the positive sign when multiplying -2 by -8. This results in -2y + 16 + 3y = 4y - 4 - y + 10. Combining like terms simplifies the equation to y + 16 = 3y + 6. Finally, isolating the variable by subtracting y and 6 from both sides gives 10 = 2y, which divides cleanly to 5.

Step-by-Step Solution

1
Distribute the outside coefficients into their respective parentheses.
−2y+16+3y=4y−4−y+10 -2y + 16 + 3y = 4y - 4 - y + 10
Distribution eliminates the parentheses, ensuring careful attention to multiplying the negative coefficient (−2×−8=+16-2 \times -8 = +16).
2
Combine the like variable terms and constant terms on each side independently.
y+16=3y+6 y + 16 = 3y + 6
Simplifying each side makes it easier to track and isolate the variable in the next steps.
3
Isolate the variable terms on one side by subtracting yy from both sides.
16=2y+6 16 = 2y + 6
Gathering all variable terms on a single side is required to solve for the unknown.
4
Isolate the constant terms by subtracting 6 from both sides, then divide by 2.
10=2y⇒y=5 10 = 2y \Rightarrow y = 5
Using inverse operations isolates yy to reveal its final numerical value.

Key Concept

Solving One-Variable Linear Equations
Question 2Question →
If 3(p+4)2−7=2p−4 \frac{3(p + 4)}{2} - 7 = 2p - 4 what is the value of pp?
Show answer & explanation

Answer: 6

Answer

6
The correct answer is found by first isolating the fractional term by adding 7 to both sides, then multiplying the entire equation by 2 to eliminate the denominator, distributing properly, and finally moving variables to one side and constants to the other to solve for p.

Step-by-Step Solution

1
Add 7 to both sides of the equation.
3(p+4)2=2p+3 \frac{3(p + 4)}{2} = 2p + 3
This isolates the fractional term on the left side of the equation.
2
Multiply both sides by 2.
3(p+4)=4p+6 3(p + 4) = 4p + 6
This clears the denominator from the fraction.
3
Distribute the 3 on the left side.
3p+12=4p+6 3p + 12 = 4p + 6
This expands the parentheses so like terms can be combined.
4
Subtract 3p from both sides.
12=p+6 12 = p + 6
This groups all variable terms on the right side of the equation.
5
Subtract 6 from both sides.
p=6 p = 6
This isolates the variable p to find its final value.

Key Concept

Solving Linear Equations with Fractions
Question 3Question →
Calculate the exact value of nn that balances the following equation:
23(6n−9)+4=3(n−2)−n \frac{2}{3}(6n - 9) + 4 = 3(n - 2) - n
Show answer & explanation

Answer: -2

Answer

The correct value for n is -2.
The solution is found by correctly applying the distributive property to both sides of the equation, combining like terms to simplify the expression to 4n−2=2n−64n - 2 = 2n - 6, and then isolating the variable nn using inverse operations to get n=−2n = -2.

Step-by-Step Solution

1
Distribute the fraction on the left side of the equation.
4n - 6 + 4 = 3(n - 2) - n
To eliminate the parentheses on the left side and isolate the individual terms.
2
Distribute the coefficient on the right side of the equation.
4n - 6 + 4 = 3n - 6 - n
To expand the expressions before combining them.
3
Combine like terms on both sides.
4n - 2 = 2n - 6
To simplify the equation into the standard linear form ax + b = cx + d.
4
Group the variable terms on one side and the constants on the other.
2n = -4
By subtracting 2n and adding 2 to both sides, we isolate the variable term.
5
Divide by the coefficient of n.
n = -2
To solve for the isolated variable n.

Key Concept

Solving multi-step linear equations involving fractions, distribution, and combining like terms.
Question 4Question →

Determine the value of hh that satisfies the equation below:

5−2(3h−4)=6h−4(h+4)−3 5 - 2(3h - 4) = 6h - 4(h + 4) - 3
Show answer & explanation

Answer: 4

Answer

The correct value is 4.
The equation is solved by first distributing the coefficients across the parentheses, which yields 5−6h+8=6h−4h−16−35 - 6h + 8 = 6h - 4h - 16 - 3. Combining like terms on both sides simplifies the equation to 13−6h=2h−1913 - 6h = 2h - 19. Adding 6h6h and 1919 to both sides isolates the variable, resulting in 32=8h32 = 8h. Dividing by 88 gives the final answer of 44.

Step-by-Step Solution

1
Apply the distributive property.
5−6h+8=6h−4h−16−35 - 6h + 8 = 6h - 4h - 16 - 3
Multiply the terms inside the parentheses by their outside coefficients, carefully distributing the negative signs.
2
Combine like terms on the left side.
13−6h=6h−4h−16−313 - 6h = 6h - 4h - 16 - 3
Combine the constant terms 55 and 88.
3
Combine like terms on the right side.
13−6h=2h−1913 - 6h = 2h - 19
Combine the variable terms (6h−4h=2h6h - 4h = 2h) and the constant terms (−16−3=−19-16 - 3 = -19).
4
Isolate the variable hh.
13+19=2h+6h13 + 19 = 2h + 6h
Add 6h6h to both sides and add 1919 to both sides to group the variables and constants.
5
Solve for the final value.
32=8h→h=432 = 8h \rightarrow h = 4
Simplify the addition and divide by 88 to find hh.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:1m 30s
Question 5Question →

What is the exact numerical value of the variable pp that balances the given algebraic equation?

Fill in the blanks below

−2(3p−5)+4p=3(p+5)−20-2(3p - 5) + 4p = 3(p + 5) - 20

Value:
Show answer & explanation

Answer

3
The correct answer is found by first distributing the outer values into the parentheses, taking special care to distribute the negative sign accurately so that −2(−5)=10-2(-5) = 10. After expanding, the equation becomes −6p+10+4p=3p+15−20-6p + 10 + 4p = 3p + 15 - 20. Combining the like terms on each side reduces the equation to −2p+10=3p−5-2p + 10 = 3p - 5. Adding 2p2p to both sides and adding 55 to both sides isolates the terms, resulting in 15=5p15 = 5p. Finally, dividing both sides by 5 yields a value of 3.

Step-by-Step Solution

1
Distribute the constants into the parentheses on both sides of the equation.
−6p+10+4p=3p+15−20-6p + 10 + 4p = 3p + 15 - 20
Expanding the grouped terms is necessary to access and combine all the variable and constant terms.
2
Combine the like terms on each side of the equation separately.
−2p+10=3p−5-2p + 10 = 3p - 5
Simplifying each side independently reduces the complexity before moving variables across the equals sign.
3
Isolate the variable by moving all pp terms to one side and all constants to the opposite side.
10+5=3p+2p10 + 5 = 3p + 2p, which simplifies to 15=5p15 = 5p
Adding 2p2p and adding 55 to both sides gathers the unknown variable exclusively to the right and constants to the left.
4
Divide by the coefficient of the variable to solve for pp.
p=3p = 3
Dividing both sides by 5 leaves the variable pp completely isolated, providing the final solution.

Key Concept

Applying the distributive property and balancing operations to solve multi-step linear equations.
Question 6Question →

Consider the following algebraic equation:

5(2m−1)−3m=2(m+10) 5(2m - 1) - 3m = 2(m + 10)

What is the value of the variable mm?

Show answer & explanation

Answer: 5

Answer

5
By correctly applying the distributive property on both sides of the equation, combining the resulting like terms, and applying algebraic operations to isolate the variable, the equation simplifies to 5m=255m = 25, which yields exactly 55.

Step-by-Step Solution

1
Distribute the constants outside the parentheses into the terms inside.
10m−5−3m=2m+2010m - 5 - 3m = 2m + 20
This removes the parentheses and allows us to combine like terms.
2
Combine the like variable terms on the left side (10m10m and −3m-3m).
7m−5=2m+207m - 5 = 2m + 20
Simplifying each side independently makes the equation easier to solve.
3
Subtract 2m2m from both sides and add 55 to both sides.
5m=255m = 25
This groups all variable terms on the left and all constant terms on the right.
4
Divide both sides by 55.
m=5m = 5
This isolates the variable mm to give the final solution.

Key Concept

Solving one-variable linear equations requiring distribution and combining like terms
Estimated Time:1m 0s
Question 7Question →

Solve the following linear equation for kk:

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

What is the exact value of kk?

Show answer & explanation

Answer: 66

Answer

The correct value is 66.
The equation is correctly solved by first using the distributive property on both sides to get 3−8k+10=−5k−53 - 8k + 10 = -5k - 5. Combining the constants on the left side yields 13−8k=−5k−513 - 8k = -5k - 5. Adding 8k8k to both sides gives 13=3k−513 = 3k - 5. Finally, adding 55 to both sides results in 18=3k18 = 3k, and dividing by 33 gives the solution k=6k = 6.

Step-by-Step Solution

1
Distribute the coefficients into the parentheses on both sides of the equation.
3−8k+10=−5k−53 - 8k + 10 = -5k - 5
To eliminate the parentheses and expose all terms for simplification.
2
Combine the constant terms on the left side of the equation.
13−8k=−5k−513 - 8k = -5k - 5
To simplify the left side.
3
Add 8k8k to both sides of the equation.
13=3k−513 = 3k - 5
To gather all terms with the variable kk on the right side.
4
Add 55 to both sides of the equation.
18=3k18 = 3k
To isolate the variable term.
5
Divide both sides by 33.
k=6k = 6
To solve for kk.

Key Concept

Solving one-variable linear equations by applying the distributive property and combining like terms.
Question 8Question →

If 5m+12=2m+275m + 12 = 2m + 27, what is the value of mm?

Show answer & explanation

Answer: 5

Answer

The correct value of the variable is 5.
By using inverse operations to collect the variable terms on one side and the constant terms on the other, the equation simplifies to 3m=153m = 15, which directly yields a result of 55.

Step-by-Step Solution

1
Move all terms containing the variable to one side of the equation.
3m+12=273m + 12 = 27
Subtracting 2m2m from both sides eliminates the variable from the right side and groups it on the left.
2
Move all constant terms to the opposite side of the equation.
3m=153m = 15
Subtracting 1212 from both sides isolates the term containing the variable.
3
Solve for the variable.
m=5m = 5
Dividing both sides by the coefficient 33 provides the final value of mm.

Key Concept

Solving Linear Equations with Variables on Both Sides
Estimated Time:45s
Question 9Question →

Solve the following linear equation for xx:

23(6x−9)−12(4x+10)=−14(8−12x)+3\frac{2}{3}(6x - 9) - \frac{1}{2}(4x + 10) = -\frac{1}{4}(8 - 12x) + 3

Fill in the blanks below

The value of xx that satisfies the equation is .
Show answer & explanation

Answer

The correct value for xx is -12.
The correct answer is derived by methodically applying the distributive property to all grouped terms on both sides of the equation. By taking special care with negative signs during the multiplication of −12-\frac{1}{2} and −14-\frac{1}{4}, the equation simplifies cleanly to 2x−11=3x+12x - 11 = 3x + 1. From there, standard algebraic isolation steps lead directly to x=−12x = -12.

Step-by-Step Solution

1
Distribute the fractional coefficients on the left side of the equation.
4x−6−2x−54x - 6 - 2x - 5
Applying the distributive property multiplies 23\frac{2}{3} by both 6x6x and −9-9, and −12-\frac{1}{2} by both 4x4x and 1010. Careful attention must be given to the negative sign.
2
Combine like terms on the left side.
2x−112x - 11
Grouping the variable terms (4x−2x4x - 2x) and constant terms (−6−5-6 - 5) simplifies the expression.
3
Distribute the fractional coefficient on the right side of the equation.
−2+3x+3-2 + 3x + 3
Multiplying −14-\frac{1}{4} by 88 gives −2-2, and −14-\frac{1}{4} by −12x-12x gives +3x+3x. The negative times negative results in a positive term.
4
Combine like terms on the right side.
3x+13x + 1
Adding the constant terms (−2+3-2 + 3) simplifies the right side.
5
Set the simplified left side equal to the simplified right side and isolate the variable xx.
x=−12x = -12
Starting with 2x−11=3x+12x - 11 = 3x + 1, subtract 2x2x from both sides to get −11=x+1-11 = x + 1. Then, subtract 11 from both sides to solve for xx.

Key Concept

Solving complex one-variable linear equations requiring careful fractional distribution and negative sign rules.
Question 10Question →

What is the value of xx in the equation below?

3−25(10x−15)=12(4x+8)−7x3 - \frac{2}{5}(10x - 15) = \frac{1}{2}(4x + 8) - 7x
Show answer & explanation

Answer: -5

Answer

-5
The correct answer is -5. By distributing the fractional coefficients, combining like terms, and isolating the variable, the equation simplifies to -4x + 9 = -5x + 4, which yields x = -5.

Step-by-Step Solution

1
Distribute the fractional coefficients on both sides of the equation.
3−4x+6=2x+4−7x3 - 4x + 6 = 2x + 4 - 7x
To eliminate the parentheses and prepare the equation for combining like terms. Carefully manage the negative sign when distributing -2/5.
2
Combine the constant terms on the left side and the variable terms on the right side.
−4x+9=−5x+4-4x + 9 = -5x + 4
To simplify both sides of the equation before isolating the variable.
3
Add 5x to both sides of the equation.
x+9=4x + 9 = 4
To keep the equation balanced while moving all x terms to one side.
4
Subtract 9 from both sides of the equation.
x=−5x = -5
To isolate the variable x and find its final value.

Key Concept

Solving multi-step linear equations involving fractions, distribution, and variables on both sides.
Question 11Question →

Determine the numerical value of the variable xx that satisfies the algebraic equation below.

Fill in the blanks below

38(16x−24)+5=−2(2x−1)+4x+6 \frac{3}{8}(16x - 24) + 5 = -2(2x - 1) + 4x + 6

x=x =
Show answer & explanation

Answer

The value of x that satisfies the equation is 2.
By successfully applying the distributive property to both sides of the equation, the expression simplifies to 6x - 9 + 5 = -4x + 2 + 4x + 6. Combining like terms reduces this to 6x - 4 = 8. Adding 4 to both sides yields 6x = 12, which gives exactly 2.

Step-by-Step Solution

1
Distribute the fraction on the left side and the negative integer on the right side.
6x−9+5=−4x+2+4x+6 6x - 9 + 5 = -4x + 2 + 4x + 6
Applying the distributive property removes the grouping parentheses, which is necessary before combining any terms.
2
Combine the constant terms on the left side and the like terms on the right side.
6x−4=8 6x - 4 = 8
Simplifying both sides of the equation by grouping like terms makes it straightforward to isolate the variable. The -4x and +4x cancel each other out.
3
Isolate the term containing the variable by adding 4 to both sides of the equation.
6x=12 6x = 12
Using the addition property of equality maintains the balance while moving constants to one side.
4
Divide both sides by 6 to solve for x.
x=2 x = 2
Dividing by the coefficient of x isolates the variable completely and provides the final solution.

Key Concept

Solving multi-step linear equations involving fractional distribution and combining like terms across both sides of the equals sign.
Question 12Question →

An algebraic relationship is given by the equation:

35(10k−25)−14(8−16k)=12(14k−10)−2(k−9) \frac{3}{5}(10k - 25) - \frac{1}{4}(8 - 16k) = \frac{1}{2}(14k - 10) - 2(k - 9)

What value of kk balances the given equation?

Show answer & explanation

Answer: 6

Answer

The value that satisfies the equation is 6.
The correct value is found by carefully distributing the fractional multipliers and negative signs to eliminate all parentheses, simplifying both sides to get 10k−17=5k+1310k - 17 = 5k + 13, and finally isolating kk by performing balanced operations to find k=6k = 6.

Step-by-Step Solution

1
Distribute the fractional coefficients into the parentheses on both sides of the equation.
(6k−15)−(2−4k)=(7k−5)−2(k−9) (6k - 15) - (2 - 4k) = (7k - 5) - 2(k - 9)
Distribution eliminates the parentheses and clears the fractional multipliers, allowing the terms to be evaluated and combined.
2
Distribute the negative signs for the remaining terms in the equation.
6k−15−2+4k=7k−5−2k+18 6k - 15 - 2 + 4k = 7k - 5 - 2k + 18
Properly applying the negative sign ensures all terms are correctly added or subtracted, especially noting that −14(−16k)=+4k- \frac{1}{4}(-16k) = +4k and −2(−9)=+18-2(-9) = +18.
3
Combine the constant terms and the variable terms independently on each side.
10k−17=5k+13 10k - 17 = 5k + 13
Simplifying each side makes it easier to isolate the variable.
4
Isolate the variable term by moving all terms containing k to one side and constants to the other side.
10k−5k=13+17⇒5k=30⇒k=6 10k - 5k = 13 + 17 \Rightarrow 5k = 30 \Rightarrow k = 6
Solving for k provides the final numerical value that balances the equation.

Key Concept

Solving complex one-variable linear equations by distributing fractional coefficients, managing negative signs, and combining like terms.
Question 13Question →

Which of the following represents the value of pp that satisfies the given equation?

−2(4p−5)+13(15p−12)=2p−(7p−14)-2(4p - 5) + \frac{1}{3}(15p - 12) = 2p - (7p - 14)
Show answer & explanation

Answer: 4

Answer

4
The correct answer is 4. To solve the equation, first distribute the outer terms across the parentheses on both sides: −2(4p−5)-2(4p - 5) becomes −8p+10-8p + 10, 13(15p−12)\frac{1}{3}(15p - 12) becomes 5p−45p - 4, and −(7p−14)-(7p - 14) becomes −7p+14-7p + 14. Combining like terms on the left side (−8p+5p-8p + 5p and 10−410 - 4) gives −3p+6-3p + 6. On the right side, combining 2p2p and −7p-7p gives −5p+14-5p + 14. Setting these equal gives −3p+6=−5p+14-3p + 6 = -5p + 14. Adding 5p5p to both sides yields 2p+6=142p + 6 = 14, and subtracting 6 from both sides gives 2p=82p = 8. Dividing by 2 results in p=4p = 4.

Step-by-Step Solution

1
Distribute the coefficients to remove parentheses on the left side.
−8p+10+5p−4=2p−(7p−14)-8p + 10 + 5p - 4 = 2p - (7p - 14)
Applying the distributive property is necessary to eliminate grouping symbols.
2
Distribute the negative sign to remove parentheses on the right side.
−8p+10+5p−4=2p−7p+14-8p + 10 + 5p - 4 = 2p - 7p + 14
Distributing the subtraction over the group allows all terms to be evaluated.
3
Combine like terms on each side of the equation independently.
−3p+6=−5p+14-3p + 6 = -5p + 14
Simplifying both sides makes it easier to balance the equation and isolate the variable.
4
Add 5p5p to both sides and subtract 6 from both sides.
2p=82p = 8
Performing these inverse operations groups all variable terms on one side and all constant terms on the other.
5
Divide both sides by 2.
p=4p = 4
Dividing by the coefficient isolates the variable pp, revealing its final value.

Key Concept

Solving complex multi-step linear equations using the distributive property and combining like terms.
Question 14Question →

A linear equation is structured as follows:

12(4y−6)−23(9−6y)=2y+15 \frac{1}{2}(4y - 6) - \frac{2}{3}(9 - 6y) = 2y + 15

Based on this relationship, what is the correct value of yy?

Show answer & explanation

Answer: 6

Answer

6
The correct value is found by properly distributing the fractional coefficients, carefully tracking negative signs, and preserving equality to isolate the variable. The operations simplify the initial equation to 6y - 9 = 2y + 15, which reduces to 4y = 24, resulting in 6.

Step-by-Step Solution

1
Distribute 12 \frac{1}{2} into the first set of parentheses.
2y−3 2y - 3
Applying the distributive property eliminates the first group.
2
Distribute −23 -\frac{2}{3} into the second set of parentheses.
−6+4y -6 + 4y
Multiplying the negative fraction across the terms. Notice that multiplying two negative values (−2/3-2/3 and −6y-6y) yields a positive result (+4y+4y).
3
Combine the distributed terms on the left side of the equation.
(2y+4y)+(−3−6)=6y−9 (2y + 4y) + (-3 - 6) = 6y - 9
Combining like terms simplifies the left side of the equation before attempting to isolate the variable.
4
Set the simplified left side equal to the right side and solve for y.
6y−9=2y+15⇒4y−9=15⇒4y=24⇒y=6 6y - 9 = 2y + 15 \Rightarrow 4y - 9 = 15 \Rightarrow 4y = 24 \Rightarrow y = 6
Using inverse operations groups the variable terms on one side and constant terms on the other to reveal the final value.

Key Concept

Solving complex linear equations involving fractional distribution and sign management.
Question 15Question →

Which value of mm is the solution to the equation below?

6m−12=2m+166m - 12 = 2m + 16
Show answer & explanation

Answer: 77

Answer

The correct solution is 77.
By subtracting 2m2m from both sides and then adding 12 to both sides, the equation simplifies to 4m=284m = 28. Dividing by 4 gives the correct result of 77.

Step-by-Step Solution

1
Subtract 2m2m from both sides of the equation to group the variable terms on the left side.
4m−12=164m - 12 = 16
This keeps all instances of the unknown variable mm on one side.
2
Add 12 to both sides of the equation to isolate the variable term.
4m=284m = 28
This cancels out the constant on the left side, leaving only the term with mm.
3
Divide both sides by 4 to solve for mm.
m=7m = 7
This removes the coefficient, revealing the final value of the variable.

Key Concept

Solving One-Variable Linear Equations
Estimated Time:45s
Question 16Question →

Calculate the value of the unknown variable.

Fill in the blanks below

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

34(8k−12)−25(15−5k)=3(k+2)+4 \frac{3}{4}(8k - 12) - \frac{2}{5}(15 - 5k) = 3(k + 2) + 4

k=k =
Show answer & explanation

Answer

5
The correct value is 5. By distributing the fractions on the left, the expression becomes 6k−9−6+2k6k - 9 - 6 + 2k. Combining like terms simplifies this to 8k−158k - 15. On the right side, distributing the 3 results in 3k+6+43k + 6 + 4, which simplifies to 3k+103k + 10. Equating both simplified sides gives 8k−15=3k+108k - 15 = 3k + 10. Subtracting 3k3k from both sides leaves 5k−15=105k - 15 = 10, and adding 1515 to both sides yields 5k=255k = 25. Dividing by 5 results in k=5k = 5.

Step-by-Step Solution

1
Distribute the fractional terms on the left side of the equation.
6k−9−6+2k6k - 9 - 6 + 2k
Multiplying 34\frac{3}{4} by (8k−12)(8k - 12) yields 6k−96k - 9. Multiplying −25-\frac{2}{5} by (15−5k)(15 - 5k) yields −6+2k-6 + 2k. Paying close attention to the negative sign during distribution is critical.
2
Distribute the terms on the right side of the equation.
3k+6+43k + 6 + 4
Multiplying 33 by (k+2)(k + 2) expands the expression, setting up the right side for simplification.
3
Combine like terms on both sides of the equal sign.
8k−15=3k+108k - 15 = 3k + 10
On the left side, combine the variable terms (6k+2k=8k6k + 2k = 8k) and constant terms (−9−6=−15-9 - 6 = -15). On the right side, combine the constant terms (6+4=106 + 4 = 10).
4
Isolate the variable kk to find its value.
5k=255k = 25, which simplifies to k=5k = 5.
Subtract 3k3k from both sides to group the variables on the left (5k−15=105k - 15 = 10). Then, add 1515 to both sides to group the constants on the right (5k=255k = 25). Dividing both sides by 55 yields the final solution.

Key Concept

Solving One-Variable Linear Equations
Question 17Question →

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

Fill in the blanks below

For the equation 7k−5=3k+197k - 5 = 3k + 19, the value of kk is .
Show answer & explanation

Answer

The correct value for k is 6.
To find the value of kk, we must isolate the variable on one side of the equation. By subtracting 3k3k from both sides, we get 4k−5=194k - 5 = 19. Next, adding 55 to both sides gives 4k=244k = 24. Finally, dividing both sides by 44 results in k=6k = 6.

Step-by-Step Solution

1
Subtract 3k3k from both sides of the equation.
4k−5=194k - 5 = 19
This groups all variable terms on the left side of the equation.
2
Add 55 to both sides of the equation.
4k=244k = 24
This isolates the variable term by moving the constant to the right side.
3
Divide both sides by 44.
k=6k = 6
This cancels the coefficient on the variable, yielding the final solution.

Key Concept

Solving linear equations with variables on both sides by using inverse operations to isolate the unknown.
Question 18Question →

An inventory manager calculates the total number of pallets, hh, that can be stored in a warehouse section using the following linear equation:

3h+6=453h + 6 = 45

What is the value of hh?

Show answer & explanation

Answer: 13

Answer

The correct value for the variable is 13.
The correct value is found by performing the inverse operations in the correct order. First, subtract 6 from both sides to get 3h = 39. Then, divide both sides by 3 to find that h = 13.

Step-by-Step Solution

1
Isolate the term with the variable by subtracting 6 from both sides of the equation.
3h=45−63h = 45 - 6
3h=393h = 39
This removes the constant term from the left side, keeping the equation balanced.
2
Solve for hh by dividing both sides of the equation by 3.
h=393h = \frac{39}{3}
h=13h = 13
Dividing by the coefficient of the variable isolates hh completely.

Key Concept

Solving basic two-step linear equations by using inverse operations to isolate the variable.
Question 19Question →

Solve the following equation for pp:

23(6p−15)+3p=−3(p−8)+2p−10 \frac{2}{3}(6p - 15) + 3p = -3(p - 8) + 2p - 10

What is the final value of pp?

Show answer & explanation

Answer: 3

Answer

The correct value for the variable is 3.
The correct answer is 3. Distributing the terms on both sides yields 4p−10+3p=−3p+24+2p−104p - 10 + 3p = -3p + 24 + 2p - 10. Combining like terms results in 7p−10=−p+147p - 10 = -p + 14. Adding pp to both sides and adding 10 to both sides gives 8p=248p = 24. Dividing by 8 isolates the variable, resulting in p=3p = 3.

Step-by-Step Solution

1
Distribute the coefficients to remove parentheses on both sides of the equation.
4p−10+3p=−3p+24+2p−104p - 10 + 3p = -3p + 24 + 2p - 10
To simplify the equation, all terms inside parentheses must be multiplied by their outer coefficients. Note that 23×6p=4p\frac{2}{3} \times 6p = 4p and 23×−15=−10\frac{2}{3} \times -15 = -10.
2
Combine like terms on each side of the equation independently.
7p−10=−p+147p - 10 = -p + 14
Combining variable terms and constant terms on the same side simplifies the equation structure before isolating the variable.
3
Move all variable terms to one side of the equation.
8p−10=148p - 10 = 14
Adding pp to both sides cancels the −p-p on the right side, grouping all unknowns on the left.
4
Isolate the variable by moving constants to the other side and dividing.
8p=24⇒p=38p = 24 \Rightarrow p = 3
Adding 10 to both sides moves the constant, and dividing both sides by the coefficient 8 solves for pp.

Key Concept

Solving complex one-variable linear equations requiring distribution and combining like terms.
Question 20Question →

A network engineer is configuring a load balancer to distribute traffic across two servers. The equation below models the traffic distribution, where tt represents the number of active connections:

5(2t−4)−3t=4(t+2)−(t−8)5(2t - 4) - 3t = 4(t + 2) - (t - 8)

Calculate the required number of active connections, tt, to successfully balance the server loads.

Show answer & explanation

Answer: 9

Answer

9
By successfully applying the distributive property on both sides, combining the variable and constant terms, and isolating the variable, the equation perfectly balances when the variable is exactly 9.

Step-by-Step Solution

1
Distribute the coefficients and negative signs across the parentheses on both sides of the equation.
10t−20−3t=4t+8−t+810t - 20 - 3t = 4t + 8 - t + 8
Distribution removes the parentheses, allowing individual terms to be combined.
2
Combine like terms on each side of the equation.
7t−20=3t+167t - 20 = 3t + 16
Simplifying each side individually makes it easier to balance the equation.
3
Subtract 3t3t from both sides of the equation.
4t−20=164t - 20 = 16
This groups all the variable terms on one side of the equation.
4
Add 20 to both sides and divide by 4.
4t=36  ⟹  t=94t = 36 \implies t = 9
Isolating the variable determines the final numerical solution.

Key Concept

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