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:

Continue Reading

\cos(\theta + 60^\circ) + \cos(\theta - 60^\circ) = 2 \cos\theta \cos 60^\circ
Since $\cos 60^\circ = \frac{1}{2}$, we have:
\cos\theta \cdot \frac{1}{2} = \cos\theta

🔗 Related Articles You Might Like:

📰 "Join the Pokemon Go Community Day Party—You Won’t Believe What’s Happening Live! 📰 Uncover the Secret Gift Inside Pokemon Go Community Day—Every Player Wants to Know! 📰 5-"Pokemon Go Community Day Live! Unforgettable Moments Everyone’s Been Waiting For!" 📰 Red Spider Lily A Bloom That Poisons Fascinates Demands Destruction 📰 Red Spider Lilys Secret Haunts Forbidden Gardens 📰 Red Squad Seizes Edge In Cornereyes Wide Follow Too Late 📰 Red Stripe Beer Like Never Before The Twist That No One Talks About 📰 Red Stripes Hidden Secret Shocking Beer That Blasts Your Taste Buds 📰 Red Sweater That Betrayed You With Every Silent Tear 📰 Red Sweater Under Scrutinywhy Its More Than Just A Coat 📰 Red Tail Catfish Stunned Fans With A Shocking Hunting Technique 📰 Red Velvet Cake So Rich Its Stealing Your Heartlearn The Secret In Seconds 📰 Red Wasp Shock The Bite That Starts With A Whisper 📰 Red Wasp Stings You Before You Notice Its Attacking 📰 Red White Blue Flags Are Trapsheres Why You Should Never Trust One Again 📰 Red Wine Glasses Youve Never Seenbut Youve Always Wanted 📰 Red Wine Vinegar Replacement Youll Never Believe Is As Simple 📰 Reddd Exposes A Hidden Truth That Changes Everything You Thought You Knew Forever

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: