Problem 29 - College Algebra with Modeling and Visualization

Question

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. Admission prices to a movie are \(\$ 4\) for children and \(\$ 7\) for adults. If 75 tickets were sold for \(\$ 456\) how many of each type of ticket were sold?

Step-by-Step Solution

Answer

23 children's tickets and 52 adult tickets were sold.

Step 1: Define the Variables

Let's let \( x \) represent the number of children's tickets sold and \( y \) represent the number of adult tickets sold. Our goal is to find the values of \( x \) and \( y \) that satisfy the given conditions.

Step 2: Set Up the Equations

We have two conditions: the total number of tickets sold and the total sales amount. From these, we can make two equations.1. The sum of the number of children's and adult tickets sold is 75: \[x + y = 75\]2. The sum of money from tickets sold should be \\(456. Children's tickets are \\)4 and adults' tickets are \$7: \[4x + 7y = 456\]

Step 3: Solve the System of Equations by Substitution

Solve the first equation for \( x \):\[x = 75 - y\]Substitute \( x = 75 - y \) into the second equation:\[4(75 - y) + 7y = 456\]This simplifies to:\[300 - 4y + 7y = 456\]Combine like terms:\[3y = 156\]Divide both sides by 3:\[y = 52\]

Step 4: Solve for the Other Variable

Now that we know \( y = 52 \), substitute it back into the equation \( x = 75 - y \):\[x = 75 - 52\]This gives:\[x = 23\]

Step 5: Verify the Solution

Substitute \( x = 23 \) and \( y = 52 \) into both original equations to verify:For the first equation:\[23 + 52 = 75\] which is true.For the second equation:\[4(23) + 7(52) = 92 + 364 = 456\] which is also true.The solution satisfies both equations.

Key Concepts

Understanding Systems of Equations

In algebra, a system of equations refers to a set of equations with multiple variables. These equations are solved together because they describe relationships between the same set of variables.
For instance, in the given problem, we have two variables: the number of children's tickets (\(x\)) and the number of adult tickets (\(y\)). Each equation in the system provides different insights into these variables:

• The first equation deals with the total number of tickets sold.
• The second equation is about the total amount of money from selling the tickets.

Solving such systems helps find the values of variables that satisfy all conditions simultaneously.

The Substitution Method Explained

The substitution method is a powerful technique to solve systems of equations. It involves substituting one variable's expression in terms of another to simplify the problem.
Here’s a step-by-step guide of how it worked in the problem:

• First, we solved one of the equations for a single variable. In this case, the first equation \(x + y = 75\) was rearranged to \(x = 75 - y\).
• This expression, \(75 - y\), replaces \(x\) in the second equation, reducing it to an equation with only one variable.
• After substitution, solving the resultant equation yields the value of one variable.

The substitution method is especially useful when one equation can easily be solved for one variable. It simplifies systems by focusing on a single variable at a time.

Variable Representation in Equations

Variable representation is crucial to understanding the problem context and translating it into mathematical language. Defining what each variable stands for sets the stage for solving an equation system.
In this exercise:

• \(x\) represents the number of children's tickets sold.
• \(y\) represents the number of adult tickets sold.

Assigning these representations allows the equation to reflect the real-world scenario effectively. This makes interpreting and solving equations more intuitive, as each variable has a clear and specific meaning tied to the problem.

Solving the Equations

The ultimate goal of dealing with systems of equations is to find values for variables that meet all conditions specified in the equations.
In this task:

• Once \(y\) was found to be 52 from the substitution, we used this value to find \(x\) by substituting back into \(x = 75 - y\). This gave \(x = 23\).
• Verification was performed by substituting \(x = 23\) and \(y = 52\) back into both original equations.
  • The first equation checked the total count of tickets: \(23 + 52 = 75\).
  • The second equation confirmed the financial tally: \(4(23) + 7(52) = 456\).

Both checks confirm the solution is correct. Solving doesn’t just find values; it includes verifying that the solution works in all scenarios given by the problem.