- Begin with the original linear equation: \(3b - 39 = -78\).
- Isolate the term containing \(b\) by adding 39 to both sides: \(3b = -39\).
- Solve for \(b\) by dividing by 3: \(b = -13\).
- The final value is \(\boxed{-13}\).
Explanation
- Problem Analysis
The problem requires solving the linear equation \(3b - 39 = -78\) for the variable \(b\). A linear equation is an equation that can be written in the form \(ax + c = d\), where \(a\), \(c\), and \(d\) are constants. Our objective is to isolate the variable \(b\) on one side of the equation to determine its value.
- Isolating the Variable Term
To isolate the variable \(b\), we first need to eliminate the constant term on the side of the equation containing \(b\). We do this by adding \(39\) to both sides of the equation:
\[3b - 39 + 39 = -78 + 39\]
\[3b = -39\]
- Solving for b
Now that we have the term \(3b\) isolated, we can solve for \(b\) by dividing both sides of the equation by the coefficient of \(b\), which is \(3\):
\[\frac{3b}{3} = \frac{-39}{3}\]
\[b = -13\]
Examples
Suppose you are keeping track of your budget for a subscription service. If you had an initial balance of \(\$39\) debt, and you made three equal payments of \(\$b\), and your final balance was \(\$78\) in debt, you could set up the equation \(3b - 39 = -78\) to determine how much each payment was. Solving this tells you each payment was \(\$13\), helping you verify your account records.