- Add 32 to both sides of the equation: \(-32 + 10b + 32 = -22 + 32\), which simplifies to \(10b = 10\).
- Divide both sides by 10 to solve for \(b\): \(b = \frac{10}{10} = 1\).
- Final answer: \(\boxed{1}\).
Explanation
- Problem Analysis
The given equation is \(-32 + 10b = -22\). This is a linear equation in one variable, \(b\). Our goal is to isolate \(b\) to determine its numerical value. We will perform algebraic manipulations on both sides of the equation to simplify it until the variable is isolated.
- Isolating the variable term
To isolate the term with \(b\), we add \(32\) to both sides of the equation:
\[-32 + 10b + 32 = -22 + 32\]
Simplifying both sides:
\[10b = 10\]
- Solving for b
Finally, we divide both sides by \(10\) to solve for \(b\):
\[\frac{10b}{10} = \frac{10}{10}\]
\[b = 1\]
Thus, the value of \(b\) is \(1\).
Examples
Solving equations is fundamental in various real-life scenarios, such as budgeting. For instance, imagine you have a debt of \(\$32\) (\(-32\)) and you take on a job that pays \(\$10\) per hour (\(10b\)). You want to find out how many hours \(b\) you need to work to reach a balance of \(\$22\) (\(-22\)). Using the equation \(-32 + 10b = -22\), we find that you need to work \(\boxed{1}\) hour to reach that specific balance.