Determine which integer in the solution set will make the equation true.
$2 p+11=19$
$S: \{-1, 1, 3, 4\}$
We are given the linear equation \(2p + 11 = 19\) and a set of candidate integer values \(S = \{-1, 1, 3, 4\}\). Our goal is to determine which value from the set \(S\) satisfies the given equation. An equation is satisfied when substituting the value into the variable \(p\) results in a true mathematical statement.
To solve for \(p\), we start by isolating the term containing \(p\) on one side of the equation. We subtract \(11\) from both sides:
\[2p + 11 - 11 = 19 - 11\]
\[2p = 8\]
Now, we divide both sides of the equation by \(2\) to solve for \(p\):
\[\frac{2p}{2} = \frac{8}{2}\]
\[p = 4\]
Thus, the solution to the equation is \(p = 4\).
We check if the calculated value \(p = 4\) exists in our provided set \(S = \{-1, 1, 3, 4\}\). Since \(4 \in S\), this value is the correct solution. Therefore, the integer that makes the equation true is \(4\), which corresponds to the last option. The final answer is \(\boxed{4}\).
Linear equations are frequently used in everyday situations involving budgeting or resource management. For example, if a store charges a fixed fee of \(\$11\) for a service plus \(\$2\) for every hour, and you have a total of \(\$19\) to spend, you can use the equation \(2p + 11 = 19\) to determine the maximum number of hours \(p\) you can afford. Solving such equations helps in making informed decisions about how to allocate limited resources effectively.