Solving Simultaneous Equations: Pencil and Eraser Cost

This article details how to solve a system of simultaneous equations by using a practical example involving pencils and erasers. Simultaneous equations are a fundamental part of algebra, and solving such problems can help understand the relationships between different variables.

The Problem Statement

The cost of a pencil and an eraser is given in two different scenarios. The first scenario states that a pencil and an eraser together cost $15. The second scenario states that two pencils and three erasers together cost $35. We need to determine the individual costs of the pencil and the eraser and then find the total cost of four pencils and nine erasers.

Solving the Simultaneous Equations

Method 1: Traditional Method

Let's denote the cost of a pencil as P and the cost of an eraser as E.

The first equation is: P E 15

The second equation is: 2P 3E 35

From the first equation, we can express E in terms of P:

E 15 - P

Substituting this into the second equation:

2P 3(15 - P) 35

2P 45 - 3P 35

45 - 35 3P - 2P

10 P

Therefore, the cost of a pencil P is $10.

Substituting P back into the first equation:

E 15 - 10 5

The cost of an eraser E is $5.

Verification

Let's verify the solution with the given conditions:

Pencil and eraser cost check: 10 5 15 (correct) Two pencils and three erasers cost check: 2 * 10 3 * 5 20 15 35 (correct)

Calculating Four Pencils and Nine Erasers

Now, let's calculate the total cost for four pencils and nine erasers:

4P 9E 4 * 10 9 * 5 40 45 85

Alternative Method

Another set of solutions attempt an alternative method:

The first set of solutions gives the cost of a pencil as $13 and an eraser as $2. The second set of solutions then calculates as follows: 2 pencils and 3 erasers: 2 * 10 3 * 5 20 15 35 4 pencils and 9 erasers: 4 * 10 9 * 5 40 45 85

Conclusion

By solving the simultaneous equations, we can determine and verify the individual costs of pencils and erasers. The solution provided here consistently matches the conditions set in the problem statement. Whether using $10 for a pencil and $5 for an eraser or adjusting to $13 for a pencil and $2 for an eraser, the final cost of four pencils and nine erasers is $85.