Dance Team Equation: Solving for Boys and Girls

A common problem in algebra involves solving equations related to a group's membership, such as the composition of a dance team. In this case, we need to determine the number of boys and girls on a dance team that has a specific total number of members. Let's explore how to write and solve the equation for this scenario.

Setting Up the Problem

Let's say a dance team has 23 members. The team has 5 more girls than boys. Our goal is to find the number of boys and girls on the team.

Variables and Equations

To solve this problem, we can use a system of equations. Let's define:

x to represent the number of boys y to represent the number of girls

We know the following:

The total number of team members is 23: ( x y 23 ) There are 5 more girls than boys: ( y x 5 )

Solving the System of Equations

To solve these equations, we can use the substitution method. Here's the step-by-step process:

Substitute the expression for y from the second equation into the first equation: ( x (x 5) 23 ) Combine like terms: ( 2x 5 23 ) Subtract 5 from both sides: ( 2x 18 ) Solve for x by dividing both sides by 2: ( x 9 )

Now that we know the number of boys is 9, we can find the number of girls:

Substitute x 9 into the second equation: ( y 9 5 14 )

Therefore, the dance team consists of 9 boys and 14 girls.

Alternative Representation

Alternatively, we can represent the problem with different variable names, such as:

b for the number of boys g for the number of girls

The equations remain the same:

( b g 23 ) ( g b 5 )

Follow the same substitution method to find the solution.

Conclusion

Understanding how to set up and solve systems of equations is crucial in algebra, and it's also applicable in many real-world scenarios, including sports and team management. By practicing these problems, you'll develop important critical thinking and problem-solving skills that are invaluable in both academics and personal life.

Key Points:

Define variables and equations based on given conditions. Use substitution to solve the system of equations. Verify the solution by plugging the values back into the original equations.