Mathematics

Example: 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. 

Solution. Let the number of boys be x and the number of girls be y. Then the equations formed are

       x + y = 10 ....(1) 

and y = x + 4 ....(2)

Let us draw the graphs of equations (1) and (2) by finding two solutions for each of the equations. The solutions of the equations are given.

x + y = 10

Plotting these points we draw the lines AB and CE passing through them to represent the equations. The two lines AB and Ce intersect at the point E (3, 7). So, x = 3 and y = 7 is the required solution of the pair of linear equations.

 
i.e. Number of boys = 3  and Number of girls = 7.

How to Verify your Answer: Putting x = 3 and y = 7 in (1), we get L.H.S. = 3 + 7 = 10 = R.H.S., (1) is verified.

Putting x = 3 and y = 7 in (2), we get 7 = 3 + 4 = 7, (2) is verified.

Hence, both the equations are satisfied.

Example: Half the perimeter of a garden, whose length is 4 more than its width is 36m. Find the dimensions of the garden.

Solution: Let the length of the garden be x and width of the garden be y. Then the equation formed are

x = y + 4 ....(1)

Half perimeter = 36

x + y = 36 ....(2)

Plotting these points we draw the lines AB and CD passing through them to represent the equations.

The two lines AB and CD intersect at the point (20, 16), So, x = 20 and y = 16 is the required solution of the pair of linear equations i.e. length of the garden is 20 m and width of the garden is 16 m.

Verify Your Answer: Putting x = 20 and y = 16 in equation no. (1), We get 20 = 16 + 4 = 20, (1) is verified.

Again, putting x = 20 and y = 16 in equation (2), we get 20 + 16 = 36 ⇒ 36 = 36, (2) is verified.

Hence, both the equations are satisfied. 

Example: Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Solution: Pair of linear equations are:

x – y + 1 = 0 ....(1)

3x + 2y – 12 = 0 ....(2)

x – y + 1 = 0

Plot the points A(0, 1), B(4, 5) and join them to get a line AB. Similarly, plot the points C(0, 6), D(2, 3) and join them to form a line CD.

Clearly, the two lines intersect each other at the point D(2, 3). Hence x = 2 and y = 3 is the solution of the given pair of equations. 

The line CD cuts the x-axis at the point E (4, 0) and the line AB cuts the x-axis at the point F(–1, 0). 

Hence, the coordinates of the vertices of the triangle are ; D(2, 3), E(4, 0), F(–1, 0).

Verify Your Answer:

Both the equations (1) and equation (2) are satisfied by x = 2 and y = 3. Hence, pair of equations are now Verified.