Solving for the Cost of Pens and Pencils: A System of Equations Approach

When dealing with cost problems, a systematic approach using mathematical equations can provide a clear and accurate solution. This article demonstrates how to use a system of equations to determine the cost of individual items, specifically pens and pencils. By following a step-by-step method, we can efficiently solve the problem presented in the title.

The Given Problem

Consider the scenario where five pens and three pencils cost $4.60, and three pens and five pencils cost $3.40. The goal is to determine the cost of a single pen, denoted by ( x ), and a single pencil, denoted by ( y ).

From the problem statement, we can write two equations:

[ 5x 3y 4.60 quad text{(Equation 1)} ]

[ 3x 5y 3.40 quad text{(Equation 2)} ]

Step-by-Step Solution

Step 1: Solve for One Variable

We begin by solving Equation 1 for ( y ):

[ 5x 3y 4.60 ]

[ 3y 4.60 - 5x ]

[ y frac{4.60 - 5x}{3} quad text{(Equation 3)} ]

Step 2: Substitute into the Second Equation

Next, we substitute Equation 3 into Equation 2:

[ 3x 5 left( frac{4.60 - 5x}{3} right) 3.40 ]

Multiplying through by 3 to eliminate the fraction:

[ 9x 5(4.60 - 5x) 10.20 ]

Distributing the 5:

[ 9x 23 - 25x 10.20 ]

Combining like terms:

[ -16x 23 10.20 ]

Step 3: Solve for ( x )

Subtract 23 from both sides:

[ -16x 10.20 - 23 ]

[ -16x -12.80 ]

Dividing by -16:

[ x frac{-12.80}{-16} 0.80 ]

This means the cost of a pen is $0.80 or 80 cents.

Step 4: Solve for ( y )

To find the cost of a pencil, substitute ( x 0.80 ) back into Equation 1:

[ 5(0.80) 3y 4.60 ]

[ 4.00 3y 4.60 ]

[ 3y 0.60 ]

[ y frac{0.60}{3} 0.20 ]

This means the cost of a pencil is $0.20 or 20 cents.

Verification

To ensure the solution is correct, we can substitute the values back into the original equations:

For Equation 1:

[ 5(0.80) 3(0.20) 4.60 ]

[ 4.00 0.60 4.60 ]

For Equation 2:

[ 3(0.80) 5(0.20) 3.40 ]

[ 2.40 1.00 3.40 ]

The values satisfy both equations, confirming our solution.

Alternative Method

An alternative method to solving the system of equations involves using a more direct approach:

First, we multiply Equation 1 by 3 and Equation 2 by 5:

[ 15P - 9PC 13.80 ]

[ 15P - 25PC 17.00 ]

Subtract Equation 1 from Equation 2:

[ 16PC 3.2 ]

[ PC 0.2 ]

Substitute ( PC 0.2 ) into Equation 1:

[ 5(0.8) - 9(0.2) 4.60 ]

[ 4.00 - 0.60 4.60 ]

[ 4.60 4.60 ]

Thus, the cost of a pencil ( P ) is $0.80 and the cost of a pen ( S ) is $0.20.

In Conclusion

Using a system of equations, we have calculated the cost of a pen and a pencil. By solving the problem step-by-step, we can ensure accuracy and reliability in our solution. This method is not only applicable to this specific problem but can be generalized to similar cost-related questions.