If the cost of 2 pencils and 3 erasers is Rs 14. Wheras the cost of 3 pencils and 5 erasers is Rs 22, then find the cost of one pencil and one eraser

Answer

To solve the problem, we need to set up a system of linear equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = cost of one pencil (in Rs)
- \( y \) = cost of one eraser (in Rs)

### Step 2: Set Up the Equations
From the problem, we have two pieces of information that can be translated into equations:

1. The cost of 2 pencils and 3 erasers is Rs 14:
\[ 2x + 3y = 14 \quad \text{(Equation 1)} \]

2. The cost of 3 pencils and 5 erasers is Rs 22:
\[ 3x + 5y = 22 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
We can solve these equations using the method of substitution or elimination. Here, we will use the elimination method.

#### Step 3.1: Multiply the Equations
To eliminate \( y \), we can multiply Equation 1 by 5 and Equation 2 by 3:

- Multiply Equation 1 by 5:
\[ 5(2x + 3y) = 5(14) \implies 10x + 15y = 70 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 3:
\[ 3(3x + 5y) = 3(22) \implies 9x + 15y = 66 \quad \text{(Equation 4)} \]

#### Step 3.2: Subtract the Equations
Now, we will subtract Equation 4 from Equation 3 to eliminate \( y \):
\[ (10x + 15y) - (9x + 15y) = 70 - 66 \]
This simplifies to:
\[ 10x - 9x = 4 \implies x = 4 \]

### Step 4: Substitute \( x \) Back to Find \( y \)
Now that we have \( x \), we can substitute it back into either Equation 1 or Equation 2 to find \( y$. We will use Equation 1:
\[ 2(4) + 3y = 14 \]
This simplifies to:
\[ 8 + 3y = 14 \]
Subtracting 8 from both sides gives:
\[ 3y = 6 \]
Dividing by 3:
\[ y = 2 \]

### Step 5: Conclusion
Thus, the cost of one pencil is Rs 4 and the cost of one eraser is Rs 2.

### Final Answer:
- Cost of one pencil (x) = Rs 4
- Cost of one eraser (y) = Rs 2

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