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:

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

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

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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Thus, the number of ways to label the 8 bird species is \(\boxed{40320}\).
Question:** A health systems analyst in Massachusetts wants to evaluate the effectiveness of 5 different health data platforms. If the analyst chooses 3 platforms to conduct a comparative study, how many different combinations of platforms can be selected?
To find the number of combinations of selecting 3 platforms out of 5, we use the combination formula:

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Final Thoughts

  • 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: