Solving Simultaneous Equations to Determine the Cost of Books and Pens

Introduction

Solving real-world problems often requires setting up and solving systems of equations. This article will walk you through a practical example where we determine the cost of each book and each pen by solving a system of two simultaneous equations. We'll also provide a step-by-step explanation and detailed calculations for clarity and educational value.

The Problem

We are given the following information: The cost of 4 books and 6 pens is $136. The cost of 6 books and 5 pens is $164. The cost of 2 books and 3 pens is $68. We need to determine the cost of one book and one pen.

Solving the System of Equations

Let's define the variables first: Let ( b ) represent the cost of one book. Let ( p ) represent the cost of one pen. Now, we can write down the equations based on the given information:

Equation 1:

4 ( b ) 6 ( p ) 136

Equation 2:

6 ( b ) 5 ( p ) 164

To solve these equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

Step 1: Decrement and Elimination

Subtract the first equation from the second equation:

(6 ( b ) 5 ( p )) - (4 ( b ) 6 ( p )) 164 - 136

2 ( b ) - 1 ( p ) 28

This simplifies to:

2 ( b ) 1 ( p ) 28

or,

2 ( b ) - 1 ( p ) 28

Step 2: Additional Information

We also have additional information: The cost of 2 books and 3 pens is $68.

2 ( b ) 3 ( p ) 68

Step 3: Solve for ( p )

Using the first simplified equation, we can find the cost of a pen.

Equation 1 (simplified):

2 ( b ) - 1 ( p ) 28

Solving for ( p ):

1 ( p ) 2 ( b ) - 28

( p ) 2 ( b ) - 28

Substitute ( p ) in the second additional equation:

2 ( b ) 3 (2 ( b ) - 28) 68

2 ( b ) 6 ( b ) - 84 68

8 ( b ) - 84 68

8 ( b ) 152

( b ) 19

Now, substitute ( b 19 ) back into the equation ( p 2 ( b ) - 28 ):

( p ) 2 × 19 - 28

( p ) 38 - 28

( p ) 10

Thus, the cost of one book is $19 and the cost of one pen is $10.

Verification

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

Original Equations:

4 ( b ) 6 ( p ) 136

6 ( b ) 5 ( p ) 164

Substituting ( b 19 ) and ( p 10 ):

4 × 19 6 × 10 76 60 136 (true)

6 × 19 5 × 10 114 50 164 (true)

Both original equations are satisfied, confirming our solution.

Conclusion

The cost of one book is $19 and the cost of one pen is $10. This solution was derived by solving a system of simultaneous equations, demonstrating the practical application of algebra in everyday problem-solving.