Question: The average of $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $ is 6. Solve for $ z $.

February 22, 2026

Solving for $ z $: Understanding the Average of Three Expressions

Understanding how to calculate the average of expressions is a key skill in algebra. One common problem students encounter is finding a variable when the average of multiple expressions is known. In this article, we’ll explore how to solve for $ z $ when the average of $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $ equals 6.

What does the average mean?

The average of three numbers is the sum divided by 3. So, if the average of $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $ is 6, we can write:

$$ rac{(4z - 3) + (2z + 5) + (z + 1)}{3} = 6 $$

Step-by-step solution

Step 1: Combine the expressions in the numerator

First, add the three expressions together:

$$ (4z - 3) + (2z + 5) + (z + 1) $$

Group like terms:

Terms with $ z $: $ 4z + 2z + z = 7z $
Constant terms: $ -3 + 5 + 1 = 3 $

So the total expression becomes:

$$ 7z + 3 $$

Now the equation looks like:

$$ rac{7z + 3}{3} = 6 $$

Step 2: Eliminate the denominator

Multiply both sides of the equation by 3:

$$ 7z + 3 = 18 $$

Step 3: Solve for $ z $

Subtract 3 from both sides:

$$ 7z = 15 $$

Divide both sides by 7:

$$ z = rac{15}{7} $$

Final Answer

$$ z = rac{15}{7} $$

Why this matters

Knowing how to set up and solve average problems helps build strong foundations in algebra. This skill is especially useful in real-world applications, such as calculating averages in data analysis, test scores, or financial planning—where finding an unknown variable based on an average is a common task.

Mastering this problem not only helps with equations but also improves logical reasoning and equation-solving confidence.

Summary

The average of $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $ is $ rac{7z + 3}{3} $
Set $ rac{7z + 3}{3} = 6 $
Solve for $ z $ to get $ z = rac{15}{7} $

Start solving algebraic word problems with clarity—understanding averages is a vital step!