Solving Cost Equations: A Practical Approach to Calculate the Price of Party Supplies

Understanding and solving cost equations is a fundamental skill in various real-life applications, from personal budgeting to business operations. One common scenario involves determining the individual and combined costs of different items. In this article, we'll explore how to calculate the cost of 8 party hats and 5 balloons using given information. We'll break down the steps and identify the individual prices of party hats and balloons.

Introduction to the Problem

The problem at hand involves two equations that represent the total cost of different quantities of party hats and balloons. Given the following equations:

5 hats 3 balloons 34 pence (Equation 1) 3 hats 2 balloons 21 pence (Equation 2)

We aim to find the cost of 8 party hats and 5 balloons by setting up and solving a system of linear equations.

Solving the System of Equations

Let's denote the cost of one party hat as h and the cost of one balloon as b. The given equations can be rephrased using these variables:

5h 3b 34 3h 2b 21

Step 1: Solve for One Variable

We start by solving the second equation for b (the cost of one balloon):

b (21 - 3h) / 2

Step 2: Substitute into the First Equation

Next, we substitute this expression into the first equation to solve for h (the cost of one party hat):

5h 3[(21 - 3h) / 2] 34

Multiplying through by 2 to clear the fraction:

10h 3(21 - 3h) 68

10h 63 - 9h 68

h 63 68

h 68 - 63

h 5

Step 3: Find the Cost of a Balloon

Now that we know h, we substitute it back into the equation for b:

b (21 - 3(5)) / 2

b (21 - 15) / 2

b 6 / 2

b 3

Step 4: Calculate the Combined Cost

With the individual costs of one party hat and one balloon known, we can now calculate the total cost for 8 party hats and 5 balloons:

8h 5b 8(5) 5(3) 40 15 55 pence

Conclusion

The cost for 8 party hats and 5 balloons is 55 pence.

Through this step-by-step process, we've successfully determined the cost of party supplies using algebraic techniques. This method can be applied to various real-world scenarios where values need to be determined from given equations.