Question 1

We will first set up equations for each problem, then find points to plot the lines and find their intersection.

Step 1 — Formulating equations for problem (i)

Let's say the number of boys is $x$. Let's say the number of girls is $y$.

The total number of students is 10. $x + y = 10 \quad \text{(Equation 1)}$

The number of girls is 4 more than the number of boys. $y = x + 4$ $y - x = 4 \quad \text{(Equation 2)}$

Step 2 — Graphing Equation 1 for problem (i)

We need some points to plot the line $x + y = 10$.

If $x = 0$: $0 + y = 10$ $y = 10$ So, we have point (0, 10).

If $y = 0$: $x + 0 = 10$ $x = 10$ So, we have point (10, 0).

If $x = 5$: $5 + y = 10$ $y = 10 - 5$

$\boxed{y = 5}$ So, we have point (5, 5).

Step 3 — Graphing Equation 2 for problem (i)

We need some points to plot the line $y - x = 4$.

If $x = 0$: $y - 0 = 4$ $y = 4$ So, we have point (0, 4).

If $y = 0$: $0 - x = 4$ $x = -4$ So, we have point (-4, 0).

If $x = 3$: $y - 3 = 4$ $y = 4 + 3$

$\boxed{y = 7}$ So, we have point (3, 7).

Step 4 — Finding the solution for problem (i)

Let's plot these points and draw the lines. The point where the two lines cross is our solution. The lines intersect at point (3, 7). This means $x = 3$ and $y = 7$.

The number of boys is 3. The number of girls is 7.

Step 5 — Formulating equations for problem (ii)

Let's say the cost of one pencil is ₹$x$. Let's say the cost of one pen is ₹$y$.

5 pencils and 7 pens cost ₹ 50. $5x + 7y = 50 \quad \text{(Equation 3)}$

7 pencils and 5 pens cost ₹ 46. $7x + 5y = 46 \quad \text{(Equation 4)}$

Step 6 — Graphing Equation 3 for problem (ii)

We need some points to plot the line $5x + 7y = 50$.

If $x = 3$: $5(3) + 7y = 50$ $15 + 7y = 50$ $7y = 50 - 15$ $7y = 35$

$\boxed{y = 5}$ So, we have point (3, 5).

If $x = 10$: $5(10) + 7y = 50$ $50 + 7y = 50$ $7y = 0$

$\boxed{y = 0}$ So, we have point (10, 0).

Step 7 — Graphing Equation 4 for problem (ii)

We need some points to plot the line $7x + 5y = 46$.

If $x = 3$: $7(3) + 5y = 46$ $21 + 5y = 46$ $5y = 46 - 21$ $5y = 25$

$\boxed{y = 5}$ So, we have point (3, 5).

If $x = 8$: $7(8) + 5y = 46$ $56 + 5y = 46$ $5y = 46 - 56$ $5y = -10$

$\boxed{y = -2}$ So, we have point (8, -2).

Step 8 — Finding the solution for problem (ii)

Let's plot these points and draw the lines. The point where the two lines cross is our solution. The lines intersect at point (3, 5). This means $x = 3$ and $y = 5$.

The cost of one pencil is ₹ 3. The cost of one pen is ₹ 5.

Answer

(i) The number of boys is 3 and the number of girls is 7. (ii) The cost of one pencil is ₹ 3 and the cost of one pen is ₹ 5.