Algebra

x + (x + 1) + (x + 2) = 42

Step-by-step solution with explanation

Final Answer

(the three consecutive integers are 13, 14, and 15)

Step-by-step solution

1

Write out the equation clearly

We have three consecutive integers starting at x. Writing them out helps us see what we need to combine.
2

Remove parentheses by grouping terms

Parentheses here don't change any signs, so we can just drop them and list all terms separately.
3

Combine like terms on the left side

We add the three x terms to get 3x, and add the constants 1 + 2 = 3.
4

Subtract 3 from both sides

To isolate the variable term, we subtract 3 from both sides of the equation.
5

Divide both sides by 3

Dividing both sides by 3 gives us the value of x. This is the only solution since this is a linear equation.
6

Verify by plugging x = 13 back in

We substitute x = 13 into the original equation. 13 + 14 + 15 = 42, which is correct!

Understanding this problem

Learning Insight

This problem models three consecutive integers, where each term is exactly 1 more than the last. Because all three terms are linear (no exponents), the equation is linear and has exactly one solution. The sum of any three consecutive integers always equals 3 times the middle number.

Quick Tip

For any three consecutive integers that sum to a number, just divide that number by 3 to get the middle integer. Here, 42 ÷ 3 = 14, the middle number, so the integers are 13, 14, 15.

Common Mistake

Students often forget to combine the constant numbers (1 and 2) separately from the x terms, writing 3x = 42 instead of 3x + 3 = 42, which gives the wrong answer of x = 14.