Question

Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) 10 students of class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
(ii) 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.

Answer 1

Hint: For part (i), assume that the number of boys is x and the number of girls is y. Now, make the equation using the information that the total number of students is 10. Then, form another equation using the information that the number of girls is 4 more than the number of boys. Now, plot the graph of the straight lines \[\left( x+y \right)=10\] and \[y=x+4\] . Now, get the coordinate of the point of the intersection of these two straight lines. Since x is the number of boys and y is the number of girls so, the x coordinate of the point of the intersection gives the number of boys and the y-coordinate of the point of the intersection gives the number of girls.

For part (ii), assume that the cost of one pencil and one pen is Rs. x and Rs. y respectively. Now, make the equation using the information that the cost of 5 pencils and 7 pens are Rs. 50. Then, form another equation using the information that the cost of 7 pencils and 5 pens are Rs. 46. Now, plot the graph of the straight lines \[5x+7y=50\] and \[7x+5y=46\] . Since x is the cost of one pencil and y is the cost of one pen, so the x coordinate of the point of the intersection gives the cost of one pencil and the y-coordinate of the point of the intersection gives the cost of one pen.

Complete step-by-step answer:
(i) First of all, let us assume that the number of boys is x and the number of girls is y.
The number of boys = x ………………………………(1)
The number of girls = y ……………………………..(2)
The total number of students = \[\left( x+y \right)\] ……………………….(3)
It is given that the total number of students is 10.
The total number of students = 10 …………………….(4)
On comparing equation (3) and equation (4), we get
\[\left( x+y \right)=10\] ………………………….(5)
From equation (1) and equation (2), we have the number of boys and girls respectively.
It is also given that the number of girls is 4 more than the number of boys.
\[y=x+4\] ……………………………..(6)
Now plotting the graph of the straight lines \[\left( x+y \right)=10\] and \[y=x+4\] .

The graph of the straight lines \[\left( x+y \right)=10\] and \[y=x+4\] are intersecting at a point A which has x-coordinate equal to 3 and y-coordinate equal to 7.
From equation (1), we have x equal to the number of boys. Since x is the number of boys so, the x coordinate of the point of the intersection gives the number of boys.
Therefore, the number of boys is 3.
Similarly, From equation (2), we have y equal to the number of girls. Since y is the number of girls so, the y coordinate of the point of the intersection gives the number of girls.
Therefore, the number of girls is 7.
Hence, the number of boys and girls are 3 and 7 respectively.
(ii) Let us assume that the cost of one pencil and one pen is Rs. x and Rs. y respectively.
The cost of one pencil = Rs. x …………………………..(1)
The cost of 5 pencils = Rs. 5x ………………………………..(2)
The cost of 7 pencils = Rs. 7x ………………………………..(3)
The cost of one pen = Rs. y …………………………..(4)
The cost of 7 pens = Rs. 7y ………………………………..(5)
The cost of 5 pens = Rs. 5y ………………………………..(6)
It is given that the cost of 5 pencils and 7 pens are Rs. 50.
Now, from equation (2) and equation (5), we get
\[5x+7y=50\] ……………………….(7)
It is given that the cost of 7 pencils and 5 pens are Rs. 46.
Now, from equation (3) and equation (6), we get
\[7x+5y=46\] ……………………….(8)
From equation (7) and equation (8), we have the equation of two straight lines.
Now, plotting the graph of the straight lines \[5x+7y=50\] and \[7x+5y=46\] .

The graph of the straight lines \[5x+7y=50\] and \[7x+5y=46\] are intersecting at a point B which has x-coordinate equal to 3 and y-coordinate equal to 5.
From equation (1), we have x equal to the cost of one pencil. Since x is the cost of one pencil so the x coordinate of the point of the intersection gives the cost of one pencil.
Therefore, the cost of one pencil is Rs. 3.
Similarly, From equation (4), we have y equal to the cost of one pen. Since y is the cost of one pen so the y coordinate of the point of the intersection gives the cost of one pen.
Therefore, the cost of one pen is Rs. 5.
Hence, the cost of one pencil and one pen is Rs. 3 and Rs. 7 respectively.

Note: We can also solve the equations of the part (i) by the substitution method.
Let us assume that the number of boys is x and the number of girls is y.
The number of boys = x.
The number of girls = y.
It is given that the total number of students is 10, \[\left( x+y \right)=10\] .
It is also given that the number of girls is 4 more than the number of boys, \[y=x+4\] .
We have the equations,
\[\left( x+y \right)=10\] …………………………..(1)
\[y=x+4\] ……………………………..(2)
Now, putting the value of y from equation (2) in equation (1), we get
\[\begin{align}
  & \left( x+y \right)=10 \\
 & \Rightarrow x+x+4=10 \\
 & \Rightarrow 2x=10-4 \\
 & \Rightarrow 2x=6 \\
\end{align}\]
\[\Rightarrow x=\dfrac{6}{2}=3\] ……………………..(2)
Now, from equation (2) and equation (3), we get
\[\begin{align}
  & \Rightarrow y=3+4 \\
 & \Rightarrow y=7 \\
\end{align}\]
Hence, the number of boys and girls are 3 and 7 respectively.