Question: The average of $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $ is 6. Solve for $ z $. - Dyverse
Solving for $ z $: Understanding the Average of Three Expressions
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.
Understanding the Context
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:
$$
rac{(4z - 3) + (2z + 5) + (z + 1)}{3} = 6
$$
Key Insights
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:
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- 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:
