Solving the Algebra Equation: x + 4x = 66 − 5x = 66 − x = 13 | Step-by-Step Guide

When faced with a math challenge like x + 4x = 66 − 5x = 66 − x = 13, clear reasoning and step-by-step breakdowns make all the difference. Whether you're a student tackling algebra for the first time or a parent helping your child understand equations, breaking down such problems helps build confidence and understanding.

Understanding the Equation

Understanding the Context

The equation presented appears in two parts:

  • Left side: x + 4x = 66
  • Right side: 66 − 5x = 66 − x = 13

Let’s simplify and solve step by step.

x + 4x = 66 → 5x = 66 → x = 13,2

Key Insights

Step 1: Simplify the Left Side

Start with the left expression:
x + 4x

Combine like terms:
x + 4x = 5x

So now the equation becomes:
5x = 66 − 5x = 66 − x = 13

Continue Reading

Equality occurs when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, which implies $ x = y = z $. Since $ x + y + z = 1 $, we have $ x = y = z = \frac{1}{3} $.
\frac{(1/3)^2}{1/3} + \frac{(1/3)^2}{1/3} + \frac{(1/3)^2}{1/3} = 3 \cdot \frac{1/9}{1/3} = 3 \cdot \frac{1}{3} = 1
Therefore, the minimum value is $ \boxed{1} $.

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Final Thoughts

Step 2: Focus on One Side — Right Side

From 66 − 5x = 66 − x = 13, notice patterns:

The phrase 66 − 5x = 66 − x suggests a way to simplify. Subtract 66 from both sides:

66 − 5x − 66 = 66 − x − 66
−5x = −x

Now divide both sides by −1:

5x = x

But this contradicts the original equation unless x = 0 — but that doesn’t satisfy 66 − x = 13, so we must interpret the equation more carefully.

Let’s simplify from 66 − 5x = 66 − x directly:
Subtract 66 from both sides:
−5x = −x

→ −5x + x = 0
→ −4x = 0
→ x = 0, again inconsistent with 66 − x = 13

So our assumption that the full expression 66 − 5x = 66 − x equals the same value may not hold unless equated properly.