Linear Equation Word Problems Worksheet - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

Problem 1

The Problem: Tickets cost $\$11$ for adults and $\$7$ for children. A group bought 8 tickets for a total of $\$76$. How many of each ticket were purchased?

Step 1: Define your variables.
Let $a$ = the number of adult tickets.
Let $c$ = the number of child tickets.

Step 2: Write the equations.
* Equation A (Total tickets): The total number of tickets is 8.
$$a + c = 8$$
* Equation B (Total cost): Adult tickets are $\$11$ and child tickets are $\$7$. The total cost is $\$76$.
$$11a + 7c = 76$$

Step 3: Solve the system.
From Equation A, we can say that $c = 8 - a$.
Substitute this into Equation B:
$$11a + 7(8 - a) = 76$$
$$11a + 56 - 7a = 76$$
Combine the 'a' terms ($11a - 7a = 4a$):
$$4a + 56 = 76$$
Subtract 56 from both sides:
$$4a = 20$$
Divide by 4:
$$a = 5$$

So, there are 5 adult tickets.
To find the child tickets, subtract 5 from the total of 8:
$$8 - 5 = 3$$
So, there are 3 child tickets.

Check: $(5 \times 11) + (3 \times 7) = 55 + 21 = 76$. Correct.

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Problem 2

The Problem: If 28 less than five times a certain number is 232, what is the number?

Step 1: Translate to an equation.
Let $x$ be the number.
"Five times a certain number" is $5x$.
"28 less than" means we subtract 28.
"is 232" means equals 232.
Equation: $5x - 28 = 232$

Step 2: Solve for x.
Add 28 to both sides:
$$5x = 232 + 28$$
$$5x = 260$$
Divide by 5:
$$x = 52$$

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Problem 3

The Problem: The sum of three consecutive even integers is 123. What are the integers?

Step 1: Analyze the math.
Consecutive even integers are numbers like 2, 4, 6 or 10, 12, 14. They are always 2 apart.
Let the first integer be $n$.
The next is $n + 2$.
The third is $n + 4$.

Equation: $n + (n + 2) + (n + 4) = 123$

Step 2: Solve.
Combine like terms:
$$3n + 6 = 123$$
Subtract 6 from both sides:
$$3n = 117$$
Divide by 3:
$$n = 39$$

Important Note: The problem asks for even integers. However, 39 is an odd number. Also, if you add any three even numbers together, the answer must be even (e.g., $2+4+6=12$). Since 123 is odd, it is impossible to get that sum using only even integers. There is likely a typo in the question (it probably meant "consecutive integers" or the sum was supposed to be 120 or 126). Assuming the question meant just "consecutive integers":
The integers would be 39, 41, and 43.

*(If strictly following the text "even integers", there is no solution).*

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Problem 4

The Problem: The sum of three consecutive even numbers is 276. What are the numbers?

Step 1: Set up the equation.
Let the first even number be $n$.
The next is $n + 2$.
The third is $n + 4$.

Equation: $n + (n + 2) + (n + 4) = 276$

Step 2: Solve.
Combine terms:
$$3n + 6 = 276$$
Subtract 6 from both sides:
$$3n = 270$$
Divide by 3:
$$n = 90$$

So the first number is 90.
The second is $90 + 2 = 92$.
The third is $90 + 4 = 94$.

Check: $90 + 92 + 94 = 276$. Correct.

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Problem 5

The Problem: The second angle of a triangle is double the first. The third angle is 40 less than the first. Find the three angles.

Step 1: Define variables.
Let the first angle be $x$.
The second angle is double the first: $2x$.
The third angle is 40 less than the first: $x - 40$.

Step 2: Use the Triangle Sum Theorem.
The angles in a triangle always add up to 180 degrees.
Equation: $x + 2x + (x - 40) = 180$

Step 3: Solve.
Combine the $x$'s ($1x + 2x + 1x = 4x$):
$$4x - 40 = 180$$
Add 40 to both sides:
$$4x = 220$$
Divide by 4:
$$x = 55$$

So, the first angle is $55^\circ$.
The second angle is $2 \times 55 = \mathbf{110^\circ}$.
The third angle is $55 - 40 = \mathbf{15^\circ}$.

Check: $55 + 110 + 15 = 180$. Correct.

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Final Answer:
1. 5 adult tickets and 3 child tickets
2. 52
3. 39, 41, 43 *(Note: The problem contains a mathematical error; these are consecutive integers, but they are not even. It is impossible to sum three even integers to get an odd number like 123.)*
4. 90, 92, 94
5. $55^\circ$, $110^\circ$, $15^\circ$