Understanding the Equation: 4 + b = 4² + 12, Simplified to 4(4 + b) = 28 – A Step-by-Step Explanation

Solving equations step-by-step is a foundational skill in algebra, and mastering them boosts confidence in working with variables. One common challenge is simplifying expressions on both sides of an equation. This article breaks down the logical progression from the initial equation to its final simplified form:
(4 + b) = 4² + 12 → 4(4 + b) = 28

Understanding the Context

Step 1: Start with the Given Equation

We begin with:
4 + b = 4² + 12

The goal is to simplify the right-hand side and solve for variable b. First, evaluate any powers and constants.

\cdot (4 + b) = 4^2 + 12 \Rightarrow 4(4 + b) = 16 + 12 = 28

Key Insights

Step 2: Evaluate the Right-Hand Side

Calculate 4² + 12:
4² = 16
So,
4² + 12 = 16 + 12 = 28

Now, replace the right-hand side:
4 + b = 28

This equation is valid but doesn’t isolate b in a simplified form involving a factored expression. To advance, we need to rewrite the equation using distributive property.

Continue Reading

Divide by 4: $ x = 24 $.
A science journalist is visualizing data on global bees. She finds that the number of monitored hives in Europe this year is 3 times that of mining hives minus 24. If there are 96 monitored hives in Europe, how many mining hives were monitored?
Let the number of mining hives be $ x $.

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Final Thoughts

Step 3: Apply the Distributive Property

The expression on the left is 4(4 + b) — you can think of (4 + b) as a parentheses that represents a single expression. To simplify, we use the distributive law:
a(b + c) = ab + ac

Here, a = 4, and the parentheses are (4 + b), so:
4(4 + b) = 4×4 + 4×b = 16 + 4b

Therefore, the original equation:
4 + b = 4² + 12
becomes:
4(4 + b) = 28

Step 4: Final Form and Verification

We now show the full simplification clearly:
Starting equation:
4 + b = 4² + 12
→
4(4 + b) = 28
since:

  • 4² + 12 = 28
  • 4(4 + b) = 16 + 4b

This simplified form makes it easier to solve for b by subtracting 4 from both sides:
b = 28 – 4 = 24

Hence, b = 24 — and verification confirms:
4 + 24 = 28, and
4² + 12 = 28, so the equation holds.