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Chapter 8: Problem 338

In the following exercises, solve each equation. $$ \frac{1}{3}(6 m+21)=m-7 $$

Short Answer

Expert verified
m = -14

Step by step solution

01

Distribute the Fraction

Apply the distributive property to \(\frac{1}{3}(6m + 21)\) to simplify it. This means to multiply \(\frac{1}{3}\) by each term inside the parentheses. So, we get: \(\frac{1}{3} \times 6m + \frac{1}{3} \times 21 = 2m + 7\). The equation now becomes: \(2m + 7 = m - 7\).
02

Move Variable Terms to One Side

Subtract \(m\) from both sides of the equation to get all variable terms on one side: \(2m - m + 7 = m - m - 7\). This simplifies to: \(m + 7 = -7\).
03

Isolate the Variable

Subtract 7 from both sides to isolate the variable \(m\): \(m + 7 - 7 = -7 - 7\). This simplifies to: \(m = -14\).
04

Verify the Solution

Substitute \(m = -14\) back into the original equation to verify the solution: \(\frac{1}{3}(6(-14) + 21) = -14 - 7\). Simplifying inside the parentheses gives \(6(-14) + 21 = -84 + 21 = -63\), so the equation becomes: \(\frac{1}{3}(-63) = -21\), and \(-14 - 7 = -21\). Both sides are equal, confirming the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a useful tool in algebra when dealing with expressions inside parentheses. In our exercise, we have the term \(\frac{1}{3}(6m + 21)\) which needs to be simplified. This property tells us to multiply \(\frac{1}{3}\) by both \(6m\) and \(21\). Therefore, \(\frac{1}{3} \times 6m + \frac{1}{3} \times 21\) simplifies to \(2m + 7\).
This step makes the equation easier to handle. Understanding the distributive property is crucial because it helps break down complex expressions into simpler components, making them easier to solve. Remember:
\(a(b+c) = ab + ac\).
Isolating Variables
Isolating the variable means getting the variable alone on one side of the equation. In our equation, after applying the distributive property, we get \(2m + 7 = m - 7\). Our goal is to isolate \(m\) on one side. First, we subtract \(m\) from both sides: \((2m - m + 7 = m - m - 7)\), which simplifies to \(m + 7 = -7\).
Isolating variables is a methodical process. We perform operations that move terms involving the variable to one side and constants to the other. This helps in finding the variable's value.
It鈥檚 like solving a puzzle step-by-step until we get the variable by itself.
Verifying Solutions
Verifying your solution is a never-skip step in algebra. After finding \(m = -14\), we substitute it back into the original equation to check our work. The original equation is \(\frac{1}{3}(6m + 21) = m - 7\).
Plugging in our solution \(m = -14\):
\(\frac{1}{3}(6(-14) + 21)\) should equal \(-14 - 7\).
Simplifying inside the parentheses, we get \(6(-14) + 21 = -84 + 21 = -63\), so we now have:
\(\frac{1}{3}(-63) = -21\), and \(-14 - 7 = -21\), and since both sides are equal, the solution is indeed correct.
Always double-check!
Equation Simplification
Simplification involves reducing the equation to its simplest form. After distributing, we had \(2m + 7 = m - 7\). To simplify, we brought similar terms together and performed operations:
\(2m - m + 7 = -7\) simplified to \(m + 7 = -7\).
Finally, by subtracting \(7\) from both sides, we simplified to \(m = -14\).
Simplifying equations step-by-step is critical as it makes the solving process straightforward and error-free. It involves carrying out arithmetic operations systematically, and ensuring every step is correct to achieve the correct solution.

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