\cdot (4 + b) = 4^2 + 12 \Rightarrow 4(4 + b) = 16 + 12 = 28
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
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.
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.
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.
Why This Problem Matters
Mastering equation simplification using properties like distribution strengthens algebraic reasoning. It’s a crucial skill for solving real-world equations, optimizing functions, and understanding more advanced math topics like systems of equations and linear algebra.
Summary
- Start with the given: 4 + b = 4² + 12
- Simplify right-hand side: 4² + 12 = 28
- Apply the distributive law: 4(4 + b) = 28
- This step-by-step transformation ensures clarity and accuracy
Whether you're a student learning algebra or a parent helping with homework, understanding how to simplify expressions and apply arithmetic rules step-by-step is invaluable.
Keywords: algebra basics, solving linear equations, distributive property, 4 + b, 4² + 12, step-by-step equation solving, algebraic simplification, solve for b, 4(4 + b) = 28.
Mastering these foundational steps paves the way for tackling complex equations with confidence. Start practicing today—simplify, apply rules, verify, and grow!
