Determine which answer in the solution set will make the equation true.
$4 f-10=3 f-18$
$S: \{-18, -8, 3, 8\}$
The equation to solve is \(4f - 10 = 3f - 18\). Our goal is to identify which value from the given set \(S = \{-18, -8, 3, 8\}\) satisfies this equation. This is a linear equation in one variable, \(f\). To solve it, we will isolate the variable \(f\) on one side of the equation.
To isolate \(f\), start by subtracting \(3f\) from both sides of the equation:
\[4f - 3f - 10 = 3f - 3f - 18\]
\[f - 10 = -18\]
Next, add \(10\) to both sides of the equation:
\[f - 10 + 10 = -18 + 10\]
\[f = -8\]
Thus, the only value of \(f\) that satisfies the equation is \(-8\).
We found that the equation holds true when \(f = -8\). Looking at our given set \(S = \{-18, -8, 3, 8\}\), we can see that \(-8\) is indeed an element of the set. Therefore, the correct answer from the provided options is \(-8\).
Solving simple linear equations is a fundamental skill used in many real-life situations. For instance, imagine you are comparing two rental car services. Company A charges a flat fee of \(\$10\) plus \(\$4\) per hour, while Company B charges \(\$18\) plus \(\$3\) per hour. You want to know at how many hours the total cost will be the same for both. This scenario leads to the equation \(4h + 10 = 3h + 18\) (or a similar variation depending on the setup), and solving for \(h\) helps you make a cost-effective decision. Understanding these algebraic steps allows you to compare options and solve optimization problems in everyday life.