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The distributive property is a useful tool when solving linear equations. It allows you to simplify an expression by 'distributing' a multiplier to each term inside parentheses. This is particularly helpful when an equation involves parentheses and a constant outside the brackets.
In our exercise, we started with \(0.5(16m + 34) = -15\). To distribute the 0.5, we multiplied it with each term inside the parentheses:
\[0.5 \times 16m + 0.5 \times 34 = -15\]
which simplifies to \[8m + 17 = -15\]
The distributive property has helped transform a complex expression into a simpler linear equation. Remember that the distributive property states: \[a(b + c) = ab + ac\]. If you follow this rule, you can handle any such expressions easily.

Isolating the variable is the method used to solve for the unknown in an equation. The goal is to get the variable by itself on one side of the equation. In our example, after simplifying using the distributive property, we had the equation:
\[8m + 17 = -15\]
To isolate \(m\), we need to perform operations that move everything else to the other side of the equation. First, we subtract 17 from both sides:
\[8m + 17 - 17 = -15 - 17\],
which simplifies to:
\[8m = -32\]
Finally, we divide both sides by 8 to solve for \(m\):
\[\frac{8m}{8} = \frac{-32}{8}\],
resulting in:
\[m = -4\]
By isolating the variable through these steps, we can determine the value of \(m\).

Linear equations are equations where the variable is raised to the power of one. These types of equations typically look like \(ax + b = c\). The solutions to linear equations like these involve straightforward algebraic manipulation.
In our exercise, we solved the linear equation:
\[0.5(16m + 34) = -15\]
Using techniques like the distributive property and isolating the variable, we transformed and simplified the equation step-by-step:
- First, apply the distributive property: \[8m + 17 = -15\]
- Then, isolate the variable by subtracting 17 from both sides: \[8m = -32\]
- Finally, solve for \(m\) by dividing by 8: \[m = -4\]
Understanding and solving linear equations is essential for many areas of mathematics. By practicing these steps, you'll become proficient in finding solutions to these types of problems.