Linear Equations Word Problems Worksheets with Answer Key - Free Printable

Let's solve each problem step by step.

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Problem 1:

Tickets to a movie cost $11 for people above 14 years (adult ticket) and $7 for people below 14 years (child ticket). A group of friends purchased 8 tickets for $76. Write equations to represent the situation. How many adult tickets and child tickets were purchased?

#### Step 1: Define variables
Let:
- $ x $ = number of adult tickets
- $ y $ = number of child tickets

#### Step 2: Set up equations
We know:
1. Total number of tickets:
$ x + y = 8 $
2. Total cost:
$ 11x + 7y = 76 $

#### Step 3: Solve the system
From equation (1):
$ y = 8 - x $

Substitute into equation (2):
$ 11x + 7(8 - x) = 76 $
$ 11x + 56 - 7x = 76 $
$ 4x + 56 = 76 $
$ 4x = 20 $
$ x = 5 $

Now find $ y $:
$ y = 8 - 5 = 3 $

✔ Answer:
- Adult tickets: 5
- Child tickets: 3

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Problem 2:

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

Let the number be $ x $.
"Five times a number" = $ 5x $
"28 less than that" = $ 5x - 28 $
Set equal to 232:

$$
5x - 28 = 232
$$

Add 28 to both sides:
$ 5x = 260 $

Divide by 5:
$ x = 52 $

✔ Answer: The number is 52

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Problem 3:

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

Wait — this is a problem!
Even integers are always even, so their sum must be even or divisible by 2. But 123 is odd, so it cannot be the sum of three even numbers.

Let’s test:
Let the integers be: $ x, x+2, x+4 $
Sum:
$ x + (x+2) + (x+4) = 3x + 6 $

Set equal to 123:
$ 3x + 6 = 123 $
$ 3x = 117 $
$ x = 39 $

But 39 is odd, so $ x = 39 $ → 39, 41, 43 — these are odd integers.

So the problem says “consecutive even integers”, but the sum is 123 (odd), which is impossible.

✘ Conclusion: There is no solution because the sum of three even integers is always even, but 123 is odd.

✔ Answer: No such even integers exist. The problem may have a typo — likely meant "consecutive odd integers".

Let’s assume it was supposed to be odd integers (since 39, 41, 43 sum to 123):

Then the integers are: 39, 41, 43

But since the question says even, we must say:

✔ Answer: No solution exists (sum of three even integers cannot be odd).

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Problem 4:

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

Let the numbers be:
$ x, x+2, x+4 $

Sum:
$ x + (x+2) + (x+4) = 3x + 6 $

Set equal to 276:
$ 3x + 6 = 276 $
$ 3x = 270 $
$ x = 90 $

So the numbers are:
90, 92, 94

Check: $ 90 + 92 + 94 = 276 $ ✔

✔ Answer: The numbers are 90, 92, 94

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Problem 5:

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

Let:
- First angle = $ x $
- Second angle = $ 2x $
- Third angle = $ x - 40 $

Sum of angles in a triangle = 180°

So:
$$
x + 2x + (x - 40) = 180
$$
$$
4x - 40 = 180
$$
$$
4x = 220
$$
$$
x = 55
$$

Now find all angles:
- First: $ x = 55^\circ $
- Second: $ 2x = 110^\circ $
- Third: $ x - 40 = 15^\circ $

Check: $ 55 + 110 + 15 = 180^\circ $ ✔

✔ Answer: The angles are 55°, 110°, and 15°

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✔ Final Answers Summary:

1. Adult tickets: 5, Child tickets: 3
2. The number is 52
3. No solution — sum of three even integers cannot be 123 (odd)
4. The numbers are 90, 92, 94
5. The angles are 55°, 110°, 15°

Let me know if you'd like these written neatly on the worksheet format!