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Chapter 15: Problem 25

Solve. $$ 8(3-m)+7=47 $$

Short Answer

Expert verified
m = -2

Step by step solution

01

- Distribute the 8

Distribute the 8 through the parentheses: \[8(3-m) + 7 = 47\]This gives: \[24 - 8m + 7 = 47\]
02

- Combine Like Terms

Combine the constant terms on the left side: \[24 + 7 - 8m = 47\]Simplify to get: \[31 - 8m = 47\]
03

- Isolate the Term with the Variable

Subtract 31 from both sides to isolate the term with the variable: \[31 - 8m - 31 = 47 - 31\]Simplifying gives: \[-8m = 16\]
04

- Solve for the Variable

Divide both sides by -8 to solve for \(m\): \[\frac{-8m}{-8} = \frac{16}{-8}\]This simplifies to: \[m = -2\]

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Key Concepts

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

Distribution in Algebra
Distribution in algebra is crucial when you have parentheses in an equation. To 'distribute' means to multiply each term inside the parentheses by the term outside. In our example, we start with:\[8(3-m) + 7 = 47\]Here, we distribute the 8 to both 3 and \(-m\). This gives us:\[24 - 8m + 7 = 47\]Distribution helps you to get rid of the parentheses and makes it easier to combine like terms later.
Combining Like Terms
Combining like terms involves adding or subtracting similar terms to simplify the equation. In the step:\[24 - 8m + 7 = 47\] '24' and '7' are both constants (numbers without variables). To combine them:\[24 + 7 - 8m = 47\]Which simplifies to:\[31 - 8m = 47\]Combining like terms simplifies the equation into fewer, easier terms to work with.
Isolating the Variable
To find the variable, we first isolate it. Start with:\[31 - 8m = 47\]We need to move the number 31 to the other side to get:\[31 - 8m - 31 = 47 - 31\]This simplifies to:\[-8m = 16\]Subtracting 31 on both sides keeps the equation balanced and starts isolating the \(m\) term.
Solving Linear Equations
The final step is to solve for the variable by performing arithmetic operations. We have:\[-8m = 16\]To find \(m\), divide both sides by -8:\[\frac{-8m}{-8} = \frac{16}{-8}\]This simplifies to:\[m = -2\]Solving linear equations often involves several steps, like distributing, combining like terms, and isolating variables, to ultimately find the value of the variable.

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