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Chapter 1: Problem 62

Solve each equation. $$(6 m+8)+(3 m+10)=90$$

Short Answer

Expert verified
The solution is \(m = 8\).

Step by step solution

01

Combine Like Terms

The given equation is \((6m + 8) + (3m + 10) = 90\). Begin by combining like terms on the left-hand side of the equation. Combine the \(m\) terms: \(6m + 3m = 9m\), and then combine the constant terms: \(8 + 10 = 18\). So, the equation becomes \(9m + 18 = 90\).
02

Isolate the Variable

Subtract 18 from both sides of the equation to isolate the term with \(m\). This gives \(9m + 18 - 18 = 90 - 18\), which simplifies to \(9m = 72\).
03

Solve for the Variable

Divide both sides of the equation by 9 to solve for \(m\). This gives \(m = \frac{72}{9}\). Simplifying the right-hand side gives \(m = 8\).

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Key Concepts

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

Combining Like Terms
When faced with a linear equation, your first task is often to simplify it. A useful strategy here is to combine like terms. Like terms are terms within the equation that have the same variable parts. In our example,
  • We have two types of terms in \((6m + 8) + (3m + 10) = 90\): terms with the variable \(m\) (like \(6m\) and \(3m\)) and constant terms (like 8 and 10).
  • To combine like terms effectively, we add the coefficients of the terms with \(m\). So, \(6m + 3m = 9m\).
  • For constant terms, we simply add them together as well, yielding \(8 + 10 = 18\).
This simplification leads us to a much cleaner equation: \(9m + 18 = 90\). By combining like terms, we've taken crucial steps towards making our equation easier to solve.
Isolate the Variable
After combining like terms, the next job is to isolate the variable. In our example, this variable is \(m\). To isolate means to get the variable alone on one side of the equation, making it easier to solve.
  • We start by removing any constants attached to the variable. In \(9m + 18 = 90\), the constant 18 is on the left side with the \(m\) term.
  • To get rid of this 18, subtract 18 from both sides of the equation: \(9m + 18 - 18 = 90 - 18\).
  • This subtraction peels away the constant, simplifying it to \(9m = 72\).
Isolating the variable helps us zero in on the solution. Once the variable is on its own, you can focus on solving it without the distraction of other numbers.
Solving for a Variable
With the variable isolated, the last step is to solve for it. Here, we're solving for \(m\) in the equation \(9m = 72\). This process involves performing arithmetic operations to find the actual value.
  • Since \(9m\) means 9 multiplied by \(m\), you'll do the opposite operation (division) to find \(m\).
  • Divide both sides of the equation by 9: \(m = \frac{72}{9}\).
  • Doing the division gives us \(m = 8\).
By solving for the variable, we've figured out the value of \(m\) that satisfies the equation. This makes everything fall into place, as you've completed the journey from a complex equation to finding the exact solution.

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