Solve for $w$.
$52=5w+12$
Simplify your answer as much as possible.
The problem asks us to solve the linear equation \(52 = 5w + 12\) for the variable \(w\). A linear equation in one variable represents a relationship where the variable is raised to the first power. To solve for \(w\), we need to isolate it on one side of the equation. Note that the information \(y = -17\) provided earlier is irrelevant to this specific equation and should be ignored.
To isolate \(w\), we first look to move the constant term on the side with the variable. Subtracting \(12\) from both sides of the equation gives:
\[52 - 12 = 5w + 12 - 12\]
\[40 = 5w\]
Now, the equation is simplified to \(5w = 40\). To find \(w\), we divide both sides by the coefficient of \(w\), which is \(5\):
\[w = \frac{40}{5}\]
\[w = 8\]
We have successfully isolated the variable \(w\) and found that \(w = 8\). We can check our result by substituting it back into the original equation:
\[5(8) + 12 = 40 + 12 = 52\]
Since this matches the original value, our solution is verified. The value of \(w\) is \(\boxed{8}\).
Linear equations are fundamental in everyday life for modeling scenarios where quantities change at a constant rate. For example, if you are saving money for a trip, and you already have \$12 in your account, and you plan to save \$5 each week, you can model your total savings after \(w\) weeks with the expression \(5w + 12\). If your goal is to reach \$52, solving the equation \(52 = 5w + 12\) will tell you exactly how many weeks it will take to reach your target. In this case, you would need to save for 8 weeks.