Question

Form the pair of linear equations in the following problem, and find their solutions graphically:

$5$ pencils and $7$ pens together cost $50$, whereas $7$ pencils and $5$ pens together cost $46$. Find the cost of one pencil and that of one pen.

Answer 1

Step 1: Assume variables and form equations

Let, the cost (in rupees) of one pencil $=x$

And, the cost (in rupees) of one pen $=y$

Then, according to the question,

Cost of $5$ pencils $+$ cost of $7$ pens $=50$

i.e., $5x+7y=50$ …(i)

And, Cost of $7$ pencils $+$ cost of $5$ pens $=46$

i.e., $7x+5y=46$ …(ii)

Step 2: Form ordered pairs

For equation (i), different values of $y$ can be obtained for the different values of $x$.

And they can be stated in the table given below,

$x$	$10$	$0$
$y$	$0$	$7.14$

Again, for equation (ii), different values of $y$ can be obtained for the different values of $x$.

And they can be stated in the table given below,

$x$	$0$	$6.57$
$y$	$9.2$	$0$

Ste 3: Plot the graph

For equation (i), we have the ordered pairs $\left(10,0\right)$ and $\left(0,7.14\right)$.

And, for equation (ii), we have the ordered pairs $\left(0,9.2\right)$ and $\left(6.57,0\right)$.

Plot these points on the same graph and joint each group of points with a straight line.

The point of intersection of the two lines so obtained is the required solution to the given problem.

From the graph, it is clear that the point of intersection of the two straight lines is $\left(3,5\right)$.

Thus, $x=3$ and $y=5$ is the solution of equation (i) and equation (ii),

So, the number of girls $=x=3$

And, the number of boys $=y=5$

Hence, the cost of one pencil and that of one pen are $\mathrm{Rs}.3$ and $\mathrm{Rs}.5$ respectively.