\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

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.

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📰 Question: What is the smallest three-digit number divisible by 11 and 13? 📰 Solution: The LCM of 11 and 13 is $143$. The smallest three-digit multiple of 143 is $143 \times 1 = 143$ (since $143 \times 2 = 286$ is also three-digit, but 143 is smaller). Thus, the answer is $\boxed{143}$.Question: Two marine biologists, Dr. Coral and Dr. Tide, arrive at a research station at random times between 7:00 and 8:00. If Dr. Coral arrives after Dr. Tide, what is the probability that Dr. Tide arrived before 7:45? 📰 Solution: Let the arrival times of Dr. Tide and Dr. Coral be $ x $ and $ y $, respectively, where $ x, y \in [0, 60] $ minutes after 7:00. The condition $ y > x $ defines a triangular region in the $ xy $-plane with area $ \frac{60^2}{2} = 1800 $. The favorable region is where $ x < 45 $ and $ y > x $. Integrating over $ x \in [0, 45] $, the area is $ \int_{0}^{45} (60 - x) \, dx = 60 \cdot 45 - \frac{45^2}{2} = 2700 - 1012.5 = 1687.5 $. The conditional probability is $ \frac{1687.5}{1800} = \frac{3}{4} $.

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.