Problem 67 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?

Step-by-Step Solution

Answer

There were 74 women and 56 men at the conference.

Step 1: Identify Variables

Let's define the variables for the problem. Let \( w \) represent the number of women and \( m \) represent the number of men attending the conference.

Step 2: Set Up the Equations

Based on the problem, we can set up two equations from the given conditions:1. The total number of faculty is 130: \( w + m = 130 \).2. There are 18 more women than men: \( w = m + 18 \).

Step 3: Substitute and Solve for One Variable

Substitute the second equation \( w = m + 18 \) into the first equation \( w + m = 130 \). This gives us:\((m + 18) + m = 130\).Simplify and solve for \( m \):\[ 2m + 18 = 130 \]Subtract 18 from both sides:\[ 2m = 112 \]Divide by 2:\[ m = 56 \].

Step 4: Solve for the Other Variable

With the value of \( m \) determined, use the equation \( w = m + 18 \) to find \( w \):\( w = 56 + 18 \)Thus, \( w = 74 \).

Step 5: Verify the Solution

Verify that the solutions satisfy both original equations:1. Check the total: \( 74 + 56 = 130 \) (correct).2. Check the difference in numbers: \( 74 = 56 + 18 \) (correct).Both conditions are satisfied, confirming our solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the unknown values that make the equation true. In the context of a system of equations, like the one in our exercise, it means finding the values of variables that satisfy all given equations simultaneously. For example, the equations from our problem are:

• \( w + m = 130 \)
• \( w = m + 18 \)

The goal is to find the pair of numbers for \( w \) (women) and \( m \) (men) that make these equations true. This requires a strategy for dealing with more than one equation at a time, as these equations are interconnected.

Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves solving one of the equations for one variable and then substituting that solution into the other equation. Here's how it worked in our exercise:First, we took the equation \( w = m + 18 \), which expresses the number of women in terms of the number of men, \( m \). This gives us a clear expression for \( w \) that we can use in another equation.We substituted \( w = m + 18 \) into the first equation \( w + m = 130 \) to get:\[(m + 18) + m = 130\]This substitution allowed us to write the system as a single equation in one variable \( m \). By solving this, we find \( m = 56 \), and subsequently, we substituted back to find \( w = 74 \). This step-by-step substitution makes solving the pair of equations more straightforward.

Identifying Variables

Identifying variables is the first and crucial step in modeling a real-world scenario as an equation. Here, we chose \( w \) for women and \( m \) for men for clarity, but any letters could be used. This identification:

• Helps set the stage for the rest of the problem-solving process.
• Represents each quantity to be determined.

By clearly assigning variables to unknown quantities, we set the foundation for writing mathematical expressions, such as \( w + m = 130 \) and \( w = m + 18 \). Identifying variables helps in organizing information and translating a word problem into algebraic equations suitable for further manipulation.

Verifying Solutions

Verifying solutions is an essential step to ensure that our found values are correct. After solving, it's crucial to check whether the pair \((w, m)\) fulfills every condition presented in the problem’s equations. To verify, you:

• Re-calculate the total faculty: \( w + m = 74 + 56 = 130 \)
• Check the difference: \( w = m + 18 \) becomes \( 74 = 56 + 18 \)

Both conditions were satisfied, confirming our solution. Verification not only reassures the solver about the correctness of the answer but also sharpens understanding as you re-evaluate the logical flow of the solution.