Solve $3 m-2+5 m=2(m-9)$.
We are given the linear equation \(3m - 2 + 5m = 2(m - 9)\). Our objective is to find the value of \(m\) that satisfies this equality. This is a linear equation in a single variable \(m\), and we can solve it by systematically simplifying both sides and isolating the variable.
First, simplify the left-hand side by combining the like terms containing \(m\). The terms are \(3m\) and \(5m\), so:
\[3m + 5m - 2 = 8m - 2\]
Next, simplify the right-hand side by distributing the constant 2 across the expression inside the parentheses:
\[2(m - 9) = 2 \cdot m - 2 \cdot 9 = 2m - 18\]
Now, we have the simplified equation:
\[8m - 2 = 2m - 18\]
To isolate \(m\), we move all terms containing \(m\) to the left side and all constant terms to the right side.
Subtract \(2m\) from both sides:
\[8m - 2m - 2 = -18\]
\[6m - 2 = -18\]
Next, add \(2\) to both sides:
\[6m = -18 + 2\]
\[6m = -16\]
Finally, solve for \(m\) by dividing both sides by the coefficient of \(m\), which is 6:
\[m = \frac{-16}{6}\]
Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, 2, we get:
\[m = -\frac{8}{3}\]
Thus, the solution to the equation is \(m = -\frac{8}{3}\).
Linear equations are fundamental in everyday life for balancing budgets, calculating travel times, or scaling recipes. For instance, if you are saving money at a constant rate and also spending a fixed amount monthly, you can set up a linear equation to determine when you will reach a specific savings goal. Solving for the variable represents finding the unique time or amount where the planned income matches the expenditure requirements, allowing for better financial planning and decision-making.