SOLVING SYSTEMS OF EQUATIONS BY GRAPHING

On this webpage solving systems of equations by graphing we are going to see how to solve linear equations by graphing method

Usually we have number of method to solve linear equations. This is one of the methods.

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.

Solution:

Total number of students in the class = 10

Let “x” be the number of boys

Let “y” be the number of girls

Number of girls is 4 more than the number of boys

y = x + 4

x – y = - 4 -----(1)

Total number of boys + number of girls = 10

                                          x + y = 10  -----(2)

                                          y = 10 – x

So far we have formed two equations from the above given information. Now we need to make a graph.

The two are intersecting at the point (3,7). Here 3 is representing the value of x and 7 is representing the value of y.

Therefore the number of boys = 3

Number of girls = 7

(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.

Solution:

Let “x” be the cost of one pencil

Let “y” be the cost of one pen

5 x + 7 y = 50 -------(1)

7 x + 5 y = 46 -------(2)

The two are intersecting at the point (3,5). Here 3 is representing the value of x and 7 is representing the value of y.

Therefore cost of one pencil = 3

Cost of one pen = 5

(2) On comparing the ratios a₁/a₂, b₁/b₂, c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

(i)  5 x – 4 y + 8 = 0

     7 x + 6 y – 9 = 0

Solution

(ii)  9 x + 3 y + 12 = 0

      18 x + 6 y + 24 = 0

Solution

(iii)  6 x - 3 y + 10 = 0

        2 x - y + 9 = 0

Solution

(3) On comparing the ratios a₁/a₂, b₁/b₂, c₁/c₂,find out whether the following pair of linear equations are consistent or inconsistent.

(i) 3 x + 2 y = 5

     2 x - 3 y = 7

Solution

(ii) 2 x - 3 y = 8

     4 x - 6 y = 9

Solution

(iii)  (3/2) x + (5/3) y = 7

      9 x - 10 y = 14

Solution

(iv)  5 x -3 y = 7

      9 x - 10 y = 14

Solution

(v) (4/3) x + 2 y = 8

      2 x + 3 y = 12

Solution

(4) Which of the following pairs of linear equations are consistent/inconsistent? if consistent,obtain the solution graphically:

(i) x + y = 5

   2 x + 2 y = 10

Solution

(ii) x - y = 8

   3 x - 3 y = 16

Solution

(iii)  2 x + y - 6 = 0

       4 x - 2 y - 4 = 0

Solution

(iv)  2 x - 2 y - 2  = 0

       4 x - 4 y  - 5 = 0

Solution

(5) Half the perimeter of a rectangular garden,whose length is 4 m more its width,is 36 m. Find the dimensions of the garden.

Solution

(6) Given the linear equation 2 x + 3 y - 8 = 0,write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) intersecting lines

(ii) Parallel lines

(iii) Coincident lines

Solution

(7) Draw the graphs of the equations x - y + 1 = 0 and 3 x + 2 y - 12 = 0. Determine the coordinates of the vertices of the triangle formed be these lines and the x-axis and shade the triangular region.

Solution




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