1
What is the value of \(n\) in the equation \(4n+3(n-5)=6\)?
Linear equations in one variable: worksheet with answers
Work through the questions first, then open the answer key and explanations to check your work.
2
A warehouse had 58,420 packages in inventory at the beginning of a month. After 14 days, 44,980 packages remained. What was the average decrease in the number of packages per day during those 14 days?
3
A warehouse had 58,320 items in inventory at the beginning of the month. After 24 days, the warehouse had 46,800 items remaining. Assuming the inventory decreased at a constant average rate, how many items did the inventory decrease per day?
4
A company starts with 3,240 packaging boxes in storage. The company uses 180 boxes each week. After how many weeks will there be 1,620 boxes remaining?
5
A water reservoir contained 94 gallons of water at the beginning of the day. Water drained from the reservoir at a constant rate of 3 gallons every 5 minutes. When the reservoir had 58 gallons remaining, how many minutes had passed?
6
A portable speaker's battery starts with 94 charge units. The battery loses 4 charge units every 5 minutes while music is playing. When the speaker has 38 charge units remaining, how many minutes has the speaker been playing music?
7
If \(8(3-2y)-11=7(3-2y)+4\), what is the value of \(3-2y\)?
8
If \(8(3-2x)-11=7(3-2x)+4\), what is the value of \(3-2x\)?
9
A bookstore has 74 journals arranged in three categories. There are \(x\) travel journals. The number of sketch journals is 5 times the number of travel journals. There are also 14 science journals. Which equation must be true for \(x\)?
10
A bookstore received a shipment of 74 items. Of the items, \(m\) are journals, the number of pens is 5 times the number of journals, and 14 of the items are calendars. Which equation must be true for \(m\)?
11
If \(\frac{-5t+9}{4}=\frac{-5t+9}{11}\), which of the following intervals contains the value of \(-5t+9\)?
12
If \(\frac{5t-9}{4}=\frac{5t-9}{11}\), which of the following intervals contains the value of \(5t-9\)?
Answer key and explanations
1. Answer
3
First distribute the 3 across the terms inside the parentheses:
\(4n+3n-15=6\)
Combine like terms:
\(7n-15=6\)
Add 15 to both sides:
\(7n=21\)
Divide both sides by 7:
\(n=3\)
Choice C is correct. A common error leading to choice A is failing to distribute the 3 to both terms inside the parentheses.
2. Answer
960
First find the total decrease in packages: \(58{,}420-44{,}980=13{,}440\).
Then divide by the number of days: \(\frac{13{,}440}{14}=960\).
So the average decrease was 960 packages per day.
3. Answer
480
First find the total decrease in inventory: \(58{,}320-46{,}800=11{,}520\).
Then divide by the number of days: \(\frac{11{,}520}{24}=480\).
So the inventory decreased by an average of 480 items per day.
4. Answer
9
Let \(x\) represent the number of weeks. The number of boxes remaining after \(x\) weeks is modeled by \(3240-180x\).
Set this expression equal to the target amount:
\(3240-180x=1620\)
Subtract 1,620 from 3,240:
\(1620=180x\)
Divide both sides by 180:
\(x=9\)
So, it will take 9 weeks for 1,620 boxes to remain.
5. Answer
60
First find how much water drained from the reservoir: \(94-58=36\) gallons.
The reservoir loses 3 gallons every 5 minutes. Since \(36 \div 3=12\), there were 12 draining intervals.
Each interval lasts 5 minutes, so the total time is \(12 \times 5=60\) minutes.
Therefore, the correct answer is 60.
6. Answer
70
First find how many charge units were used: \(94-38=56\).
The battery loses 4 charge units every 5 minutes. Since \(56\div 4=14\), there were 14 intervals of battery loss.
Each interval lasts 5 minutes, so the total time is \(14\times 5=70\) minutes.
Therefore, the correct answer is 70.
7. Answer
15
Notice that the same expression, \(3-2y\), appears on both sides of the equation. Treat it as a single quantity.
Subtract \(7(3-2y)\) from both sides:
\(8(3-2y)-11-7(3-2y)=4\)
\(3-2y-11=4\)
Add 11 to both sides:
\(3-2y=15\)
Therefore, the value of \(3-2y\) is \(15\).
8. Answer
15
Notice that the same expression, \(3-2x\), appears on both sides of the equation. Let \(E=3-2x\). The equation becomes \(8E-11=7E+4\).
Subtract \(7E\) from both sides:
\(E-11=4\)
Add 11 to both sides:
\(E=15\)
Therefore, the value of \(3-2x\) is \(15\).
9. Answer
\(6x+14=74\)
The number of travel journals is \(x\). The number of sketch journals is 5 times that amount, so it is \(5x\). The number of science journals is 14.
Since the total number of journals is 74, add all three categories:
\(x+5x+14=74\)
Combine like terms:
\(6x+14=74\)
Therefore, the correct equation is \(6x+14=74\).
10. Answer
\(6m+14=74\)
The shipment has three categories: journals, pens, and calendars. The number of journals is \(m\). The number of pens is 5 times the number of journals, so there are \(5m\) pens. There are 14 calendars.
Add the three category counts to equal the total number of items:
\(m+5m+14=74\)
Combine like terms:
\(6m+14=74\)
So the correct answer is choice C.
11. Answer
\(-2<x<3\)
Multiply both sides of the equation by 44, the least common multiple of 4 and 11:
\(11(-5t+9)=4(-5t+9)\)
Subtract \(4(-5t+9)\) from both sides:
\(7(-5t+9)=0\)
Therefore, \(-5t+9=0\). The only interval that contains 0 is \(-2<x<3\).
12. Answer
\(-2<x<3\)
Multiply both sides of the equation by 44, the least common multiple of 4 and 11:
\(11(5t-9)=4(5t-9)\)
Subtract \(4(5t-9)\) from both sides:
\(7(5t-9)=0\)
Therefore, \(5t-9=0\). The only interval that contains 0 is \(-2<x<3\).