\cdot (4 + b) = 4^2 + 12 \Rightarrow 4(4 + b) = 16 + 12 = 28 - Veritas Home Health
Understanding the Equation: 4 + b = 4² + 12, Simplified to 4(4 + b) = 28 – A Step-by-Step Explanation
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.
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.
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.
