Problem 59 - Algebra 2

Question

For Exercises \(58-63,\) define a variable, write an equation, and solve the problem. Schoot A school conference room can seat a maximum of 83 people. The principal and two counselors need to meet with the school's student athletes to discuss eligibility requirements. If each student must bring a parent with them, how many students can attend each meeting?

Step-by-Step Solution

Answer

Maximum 40 students can attend.

Step 1: Define the Variable

Let's define the variable that will help us solve the problem. Let \( x \) represent the number of students attending the meeting. Each student also brings one parent.

Step 2: Write the Equation

We are given that the principal, two counselors, students, and their parents must all fit into the room. The equation that represents the total number of people at the meeting is: \[ 1\text{ (principal)} + 2\text{ (counselors)} + x\text{ (students)} + x\text{ (parents)} \leq 83 \] This simplifies to: \[ 3 + 2x \leq 83 \]

Step 3: Solve the Equation

Now, we solve the inequality \( 3 + 2x \leq 83 \):1. Subtract 3 from both sides: \[ 2x \leq 80 \]2. Divide both sides by 2: \[ x \leq 40 \]

Step 4: Interpret the Solution

The inequality \( x \leq 40 \) means that a maximum of 40 students can attend the meeting with their parents.

Key Concepts

Variable definition

In the context of algebra, defining a variable is like setting the stage for solving a problem. Variables are symbols, typically letters, that represent unknown numbers or values in a mathematical equation. In our problem, we need to find out how many students can attend a meeting at a school conference room. Each student is required to bring a parent, which doubles the total number of attendees connected to each student.

For this purpose, we choose a variable to represent the unknown quantity we are interested in. Here, we define the variable as follows: Let \( x \) represent the number of students attending the meeting. Consequently, each student bringing one parent implies there are \( x \) students and \( x \) parents attending. Thus, the total number of participants related to students is \( 2x \). Defining variables clearly helps us develop equations that model our real-world problems.

Solving inequalities

Solving inequalities involves finding the values of variables that satisfy a given inequality condition. First, we translate the real-world context of our problem into a mathematical inequality. The principal, two counselors, students, and their parents all must fit into a room that seats a maximum of 83 people.

To express this situation mathematically, we can write the inequality as \[ 1 + 2 + x + x \leq 83 \] which simplifies as \( 3 + 2x \leq 83 \). Our goal is to find the maximum number of students \( x \) that meets the condition of this inequality.

We solve the inequality step-by-step:

• First, subtract 3 from both sides: \( 2x \leq 80 \).
• Then, divide both sides by 2 to isolate \( x \): \( x \leq 40 \).

This process identifies the largest possible number of students that can attend with their parents without exceeding the room's capacity.

Interpreting solutions

Once an inequality is solved, interpreting the solution is crucial to making sense of the answer in real-world terms. In the final step of solving \( 3 + 2x \leq 83 \), we determine that \( x \leq 40 \). What does this tell us?

The inequality result indicates the maximum number of students allowed to attend the meeting is 40. This is because each student is accompanied by a parent, doubling the number of seats occupied per student. Hence, the expression "\( x \leq 40 \)" directly translates to having a maximum of 40 students present at the meeting.

Interpreting solutions effectively involves checking if they align with any given constraints in the problem. Here, ensuring the total number does not exceed 83 aligns perfectly with the seating restriction of the conference room, making it a valid solution.