Let:

• $x$ be the cost of one pencil, and
• $y$ be the cost of one pen.

We can set up two equations based on the information provided:

1. The total cost of 5 pencils and 7 pens is Rs. 50: $5x + 7y = 50$

2. The total cost of 7 pencils and 5 pens is Rs. 46: $7x + 5y = 46$

Step 1: Solve the system of equations

To solve these equations, we can use the elimination method.

Multiply the first equation by 7 and the second equation by 5 to make the coefficients of $y$ the same:

$7 \times (5x + 7y) = 7 \times 50$ $5 \times (7x + 5y) = 5 \times 46$

This gives us:

1. $35x + 49y = 350$
2. $35x + 25y = 230$

Step 2: Eliminate $x$

Subtract the second equation from the first:

$(35x + 49y) - (35x + 25y) = 350 - 230$ $24y = 120$ $y = \frac{120}{24} = 5$

Step 3: Substitute $y = 5$ into one of the original equations

Using the first equation:

$5x + 7(5) = 50$ $5x + 35 = 50$ $5x = 15$ $x = \frac{15}{5} = 3$

Solution

The cost of one pencil is Rs. 3, and the cost of one pen is Rs. 5.

Would you like further details or have any questions about this solution?

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1. How do you use substitution to solve systems of equations?
2. What is the elimination method, and how does it work with linear equations?
3. How can you verify solutions to systems of equations?
4. What are some applications of systems of equations in real life?
5. How can systems of equations be solved graphically?

Tip: When working with systems of equations, checking your solution by substituting values back into the original equations is a great way to confirm accuracy.