Page No 3.29
Q.19. Determine graphically the vertices of the triangle, the equations of whose sides are given below :
(i) 2y − x = 8, 5y − x = 14 and y − 2x = 1
(ii) y = x, y = 0 and 3x + 3y = 10
Ans. (i) Draw the 3 lines as given by equations
By taking x=1 = 1 cm on x−axis
And y =1=1cm on y−axis
Clearly from graph points of intersection three lines are
(−4,2) , (1,3), (2,5)
(ii) Draw the 3 lines as given by equations
By taking x=1 = 1 cm on x−axis
And y =1=1cm on y−axis
y = 0
y = x
From graph point of intersection are (0,0) (10/3,0) (5/3,5/3)
Q.20. Determine, graphically whether the system of equations x − 2y = 2, 4x − 2y = 5 is consistent or in-consistent.
Ans. The given equations are
x - 2y = 2 .....(i)
4x - 2y = 5 .....(ii)
Putting x = 0 in equation (i), we get:
⇒ 0 - 2y = 2
⇒ y = -1
⇒ x = 0, y = - 1
Putting y = 0 in equation (i) we get:
⇒ x - 2 x 0 = 2
⇒ x = 2
⇒ x = 2, y = 0
Use the following table to draw the graph.
Draw the graph by plotting the two points A(0, - 1), B(2, 0) from table.
4x - 2y = 5 .....(i)
Putting x = 0 in equation (ii) we get:
⇒ 4 x 0 - 2y = 5
⇒ y = - 5/2
⇒ x = 0, y = -5/2
Putting y = 0 in equation (ii) we get:
⇒ 4x - 2 x 0 = 5
⇒ x = 5/4
⇒ x = 5/4, y = 0
Use the following table to draw the graph.
Draw the graph by plotting the two points C(0,-5/2), D(5/4,0) from table.
It has unique solution.
Hence the system of equations is consistent
Q.21. Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not :
(i) 2x − 3y = 6, x + y = 1
(ii) 2y = 4x − 6, 2x = y + 3
Ans. (i) The given equations are
2x - 3y = 6 ......(i)
x + y = 1 ......(ii)
Putting x = 0 in equation (i), we get:
⇒ 2 x 0 - 3y = 6
⇒ y = -2
x = 0, y = -2
Putting y = 0 in equation (i) we get:
⇒ 2x - 3 x 0 = 6
⇒ x = 3
x = 3, y = 0
Use the following table to draw the graph.
Draw the graph by plotting the two points A(0, - 2), B(3,0) from table.
Graph of the equation....(ii):
x + y = 1.....(ii)
Putting x = 0 in equation (ii) we get:
⇒ 0 + y = 1
⇒ y = 1
∴ x = 0, y = 1
Putting y = 0 in equation (ii) we get:
⇒ x + 0 = 1
⇒ x = 1
x = 1, y = 0
Use the following table to draw the graph.
Draw the graph by plotting the two points C(0,1),D(1,0) from table.
The two lines intersect at point
Hence the equations have unique solution.
(ii) The equations of graphs is
2y = 4x - 6
4x - 2y = 6 ....(i)
2x = y + 3
2x - y = 3 ....(ii)
Putting x = 0 in equation (i), we get:
⇒ 4 x 0 - 2y = 6
⇒ y = -3
⇒ x = 0, y = -3
Putting y = 0 in equation (i), we get:
⇒ 4x - 2 x 0 = 6
⇒ x = 3/2
x = 3/2, y = 0
Use the following table to draw the graph.
The graph of (i) can be obtained by plotting the two points A(0, - 3),B(3/2,0).
Graph of the equation (ii)
2x - y = 3 ......(ii)
Putting x = 0 in equation (ii), we get:
⇒ 2 x 0 - y = 3
⇒ y = -3
x = 0, y = -3
Putting y = 0 in equation (ii), we get:
⇒ 2x - 0 = 3
⇒ x = 3/2
x = 3/2, y = 0
Use the following table to draw the graph.
Draw the graph by plotting the two points C(0,-3),D(3/2,0) from table.
The two lines are coincident.
Hence the equations have infinitely much solution.
Hence the system is consistent.
Q.22. Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.
(i) 2x − 5y + 4 = 0,
2x + y − 8 = 0
(ii) 3x + 2y = 12,
5x − 2y = 4
(iii) 2x + y − 11 = 0,
x − y − 1 = 0
(iv) x + 2y − 7 = 0,
2x − y − 4 = 0
(v) 3x + y − 5 = 0,
2x − y − 5 = 0
(vi) 2x − y − 5 = 0,
x − y − 3 = 0
Ans. (i) The given equations are
2x - 5y + 4 = 0 .....(i)
2x + y - 8 = 0 .....(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 3, y = 2.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the pointsrespectively.
(ii) The given equations are
3x + 2y = 12 .......(i)
5x - 2y = 4 .......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 2, y = 3.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0, 6) and (0, -2) respectively.
(iii) The given equations are
2x + y - 11 = 0 ........(i)
x - y - 1 = 0 ........(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 4, y = 3.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0, 11) and (0, -1) respectively.
(iv) The given equations are
x + 2y - 7 = 0 ...... (i)
2x - y - 4 = 0 ...... (ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 3, y = 2.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points(0,3.5) and (0,-4) respectively.
(v) The given equations are
3x + y - 5 = 0 ......(i)
2x - y - 5 = 0 ......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 2, y = −1.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points(0,5) and (0, -5) respectively.
(vi) The given equations are
2x - y - 5 = 0 .......(i)
x - y - 3 = 0 .......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 2, y = −1.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0,-3) and (0,-5) respectively.
Q.22. Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.
(i) 2x − 5y + 4 = 0,
2x + y − 8 = 0
(ii) 3x + 2y = 12,
5x − 2y = 4
(iii) 2x + y − 11 = 0,
x − y − 1 = 0
(iv) x + 2y − 7 = 0,
2x − y − 4 = 0
(v) 3x + y − 5 = 0,
2x − y − 5 = 0
(vi) 2x − y − 5 = 0,
x − y − 3 = 0
Ans. (i) The given equations are
2x - 5y + 4 = 0 ......(i)
2x + y - 8 = 0 ......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 3, y = 2.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the pointsand (0,8) respectively.
(ii) The given equations are
3x + 2y = 12 .......(i)
5x - 2y = 4 .......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 2, y = 3.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0,6) and (0,-2) respectively.
(iii) The given equations are
2x + y - 11 = 0 ......(i)
x - y - 1 = 0 ......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 4, y = 3.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0, 11) and (0, - 1) respectively.
(iv) The given equations are
x + 2y - 7 = 0 ......(i)
2x - y - 4 = 0 ......(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 3, y = 2.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0,3.5) and (0, -4) respectively.
(v) The given equations are
3x + y - 5 = 0 ....(i)
2x - y - 5 = 0 ....(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 2, y = −1.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0,5) and (0, -5) respectively.
(vi) The given equations are
2x - y - 5 = 0 .....(i)
x - y - 3 = 0 .....(ii)
The two points satisfying (i) can be listed in a table as,
The two points satisfying (ii) can be listed in a table as,
Now, graph of equations (i) and (ii) can be drawn as,
It is seen that the solution of the given system of equations is given by x = 2, y = −1.
Also, it is observed that the lines (i) and (ii) meet the y-axis at the points (0,-3) and (0,-5) respectively.
respectively.
and (0,8) respectively.
