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Chapter 8: Problem 35

In the following exercises, solve each equation. $$ 2(8 m+3)-15 m-4=3-5 $$

Short Answer

Expert verified
m = -4

Step by step solution

01

Distribute the 2

First, distribute the 2 into the parentheses in the term \(2(8m + 3)\). This means that you multiply both 8m and 3 by 2:\[2(8m + 3) = 2 \times 8m + 2 \times 3 = 16m + 6\]
02

Simplify the equation

Now, substitute the distributed term back into the equation:\[16m + 6 - 15m - 4 = 3 - 5\]Combine like terms on both sides of the equation:\[ (16m - 15m) + (6 - 4) = 3 - 5\]Which simplifies to:\[ m + 2 = -2\]
03

Solve for m

To solve for m, subtract 2 from both sides of the equation:\[ m + 2 - 2 = -2 - 2\]Which simplifies to:\[ m = -4\]

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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 lets you multiply a single term by each term inside the parentheses. In our exercise, we distribute the 2 across the terms inside the parentheses:
  • First, take the number outside the parentheses (2) and multiply it by the first term inside the parentheses (8m).
  • Second, multiply 2 by the second term inside the parentheses (3).
This gives us:
  • 2 times 8m equals 16m
  • 2 times 3 equals 6
Putting it all together, we get: (16m + 6). This new expression replaces the original expression in the equation.
Combining Like Terms
Once you've distributed the terms, the next step is to combine like terms.
Like terms are terms that have the same variable part; for example, 16m and -15m are like terms because both contain the variable m.
If we look at our equation:
  • 16m - 15m are like terms and can be combined to give us 1m or simply m.
  • 6 - 4 are numerical like terms which combine to give us 2.
This simplifies the equation to: \[ m + 2 = -2 \]
Equation Simplification
Simplifying an equation involves combining all like terms and reducing complex expressions to simpler ones.
In our simplified equation \[ m + 2 = -2 \], each side has single terms that are easy to work with. This step makes it much easier to isolate the variable.
Always check to ensure you have combined all possible terms, so no complexity remains in the equation.
Solving for a Variable
The goal in solving a linear equation is to isolate the variable on one side of the equation.
In our simplified equation, \[m + 2 = -2 \], we need to find the value of m. Subtract 2 from both sides:
  • This involves -2 - 2 giving us -4 on the right-hand side.
Thus, we have:
  • m + 2 - 2 = -4
  • m = -4
This means m is equal to -4, which is the solution to the equation.

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