Solving Simultaneous Equations for Ticket Prices in a Cinema

Understanding how to solve simultaneous equations is a fundamental skill in algebra. This article will walk you through solving a real-life problem involving the pricing of cinema tickets. By the end, you will be able to calculate the price of an adult ticket and a child ticket based on given spending information.

Introduction to Simultaneous Equations

Simultaneous equations are a set of two or more equations with the same variables. These equations are often used to find the values of the variables. In this case, we are dealing with the cost of tickets at a cinema. Let's start with the problem and then break down the solution step by step.

Problem Statement

At a cinema, Sam paid ￡30 for three adult tickets and one child ticket. Similarly, he paid ￡22 for one adult ticket and three child tickets. We aim to determine the cost of each type of ticket.

Solution

Method 1: Using Algebraic Substitution

We can set up the following equations based on the given information:

Let the cost of an adult ticket be x and the cost of a child ticket be y.

From the problem, we have:

3x y 30 (Equation 1) x 3y 22 (Equation 2)

To solve these equations, we can first manipulate Equation 1 by multiplying it by 3:

9x 3y 90 (New Equation 1)

Now, subtract Equation 2 from the new Equation 1:

9x 3y - (x 3y) 90 - 22

This simplifies to:

8x 68

Divide both sides by 8:

x 8.50

Now substitute x 8.50 back into one of the original equations, let’s choose Equation 1:

3(8.50) y 30

25.50 y 30

Solving for y:

y 30 - 25.50

y 4.50

So, the cost of an adult ticket is ￡8.50 and the cost of a child ticket is ￡4.50.

Verification

Let's verify the calculations by substituting the values back into the original equations:

3(8.50) 4.50 25.50 4.50 30 8.50 3(4.50) 8.50 13.50 22

The calculations are correct, and the solution checks out.

Conclusion

In this article, we have successfully solved a practical problem using simultaneous equations. We started with the given data about the cinema ticket prices and systematically solved the equations to find the cost of adult and child tickets. Understanding such algebraic methods is invaluable in many real-world scenarios, including finance, economics, and even in everyday problem-solving.

Related Keywords

simultaneous equations, ticket pricing, algebraic problem solving