Solving a System of Equations to Determine the Cost of Pens and Pencils

In this article, we will solve a system of linear equations to determine the cost of a pen and a pencil based on the given information. The approach involves setting up and solving two equations derived from the problem statement.

Introduction to the Problem

The problem presents a scenario where the cost of 6 pens and 3 pencils is 60 units, and the cost of 5 pens and 2 pencils is 48 units. Our task is to find the individual costs of each item using algebraic methods.

Setting Up the Equations

We define:

n - p cost of one pen

c - p cost of one pencil

Based on the given information, we can set up the following system of equations:

6p 3c 60 (Equation 1)

5p 2c 48 (Equation 2)

Solving the System of Equations

Step 1: Simplifying Equation 1

We can simplify Equation 1 by dividing all terms by 3:

2p c 20 (Equation 3)

Step 2: Expressing c in Terms of p

From Equation 3, we can express c in terms of p:

c 20 - 2p (Equation 4)

Step 3: Substituting Equation 4 into Equation 2

Now, we substitute Equation 4 into Equation 2:

5p 2(20 - 2p) 48

Step 4: Simplifying and Solving for p

Expanding and simplifying the equation gives:

5p 40 - 4p 48

Combining like terms:

p 40 48

Subtracting 40 from both sides:

p 8

Step 5: Substituting p Back to Find c

Now that we have p 8, we substitute it back into Equation 4 to find c:

c 20 - 2(8) 20 - 16 4

Conclusion

The cost of each pen and pencil is:

Cost of one pen: $8

Cost of one pencil: $4

Alternatives to Solve the Problem

Alternatively, we can use different methods to solve the same problem:

Method 1: Using Different Multipliers

We can use the method of multiplying the first equation by 2 and the second equation by 3:

12p 6c 120 (Equation 5) 15p 6c 144 (Equation 6)

Subtracting Equation 5 from Equation 6 gives:

-3p -24

p 8

Substituting p 8 into Equation 1 gives:

6(8) 3c 60

48 3c 60

3c 12

c 4

Method 2: Subtracting the Multiplied Equations

We can also use the method of subtracting the equations after multiplying:

12p 6c 120 (Equation 5)

15p 6c 144 (Equation 6)

Subtracting Equation 5 from Equation 6:

-3p -24

p 8

Substituting p 8 into Equation 1:

48 3c 60

3c 12

c 4

Conclusion with Python Code

To validate our results, we can use Python to solve the equations:

import sympy as spp, c  ('p c')# Define the equationseq1  sp.Eq(6*p   3*c, 60)eq2  sp.Eq(5*p   2*c, 48)# Solve the equationssolution  ((eq1, eq2), (p, c))p_cost, c_cost  solution[p], solution[c]p_cost, c_cost

Output: (8, 4)

The calculations confirm that the cost of one pen is $8 and the cost of one pencil is $4.