What value of $m$ makes the equation true?
$16+2(5 m-8)-m=36$
The problem requires solving the linear equation \(16 + 2(5m - 8) - m = 36\) for the variable \(m\). A linear equation is an equation that can be written in the form \(ax + b = 0\), and our goal is to isolate \(m\) by performing algebraic operations on both sides of the equation while maintaining equality.
First, we expand the term \(2(5m - 8)\) using the distributive property, which states that \(a(b - c) = ab - ac\). Applying this, we get \(2 \times 5m - 2 \times 8 = 10m - 16\). Substituting this back into the original equation, we have:
\[16 + 10m - 16 - m = 36\]
Next, we combine like terms on the left side of the equation. Grouping the variable terms \(10m\) and \(-m\), we get \(10m - m = 9m\). Grouping the constant terms \(16\) and \(-16\), we get \(16 - 16 = 0\). The equation simplifies to:
\[9m + 0 = 36\]
\[9m = 36\]
To isolate \(m\), we divide both sides of the equation by the coefficient of \(m\), which is \(9\):
\[m = \frac{36}{9}\]
\[m = 4\]
Thus, the value of \(m\) that satisfies the equation is \(4\).
Solving linear equations is essential in everyday scenarios where you need to determine an unknown quantity based on known relationships. For instance, if a cellular plan charges a flat base fee plus a rate per gigabyte, you can use a linear equation to find how much data you consumed if you know your total monthly bill. By setting up the equation, you can easily calculate the unknown data usage just as we solved for \(m\).