- Set up the equation for the sum of two consecutive numbers: \(n + (n + 1) = 157\).
- Simplify the equation: \(2n + 1 = 157\).
- Solve for \(n\) by isolating the variable: \(2n = 156\), so \(n = 78\).
- Conclude that the first number is \(\boxed{78}\).
Explanation
- Understanding the Problem
The problem asks us to find the first number in a pair of consecutive numbers whose sum is \(157\). Let \(n\) represent the first number. Since consecutive numbers follow one another, the second number is \(n + 1\). The problem provides the equation representing their sum: \(n + (n + 1) = 157\), which simplifies to \(2n + 1 = 157\). Our goal is to solve this linear equation for \(n\).
- Solving the Equation
Starting with the equation \(2n + 1 = 157\), we first isolate the term with \(n\) by subtracting \(1\) from both sides:
\[2n = 157 - 1\]
\[2n = 156\]
Next, to solve for \(n\), we divide both sides by \(2\):
\[n = \frac{156}{2}\]
Performing the division, we find \(n = 78\).
- Conclusion and Verification
We have determined that \(n = 78\). To verify, if the first number is \(78\), then the consecutive number is \(78 + 1 = 79\). The sum of these two numbers is \(78 + 79 = 157\), which matches the condition given in the problem. Comparing this result with the provided options, the correct answer is \(78\).
Examples
Understanding how to model problems using equations is a fundamental skill in algebra. For instance, if you and a friend are playing a game where you collect coins and you know the total number of coins collected by two people is a specific sum, you can use these simple linear equations to determine exactly how many coins each person has. This kind of logical modeling is used in everything from personal finance to inventory management.