Question

Check graphically whether the pair of equations $3x-2y+2=0\mathrm{and}\frac{3}{2}x-y+3=0$ is consistent. Also, find the coordinates of the points, where the graphs of the equations meet the y-axis.

Answer 1

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 3x − 2y + 2 = 0

3x − 2y + 2 = 0
⇒ 2y = ( 3x + 2)
∴ $y=\frac{3x+2}{2}$ ...............(i)
Putting x = 0, we get y = 1.
Putting x = 2, we get y = 4.
Putting x = −2, we get y = − 2.
Thus, we have the following table for the equation 3x − 2y + 2 = 0.

x	0	2	−2
y	1	4	−2

Now, plot the points A(0, 1), B(2 , 4) and C(−2, −2) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 3x − 2y + 2 = 0 and it meets the y-axis at A(0, 1).

Graph of $\frac{3}{2}x-y+3=0$
$\frac{3}{2}x-y+3=0$
∴ $y=\frac{3}{2}x+3$...............(ii)
Putting x = 0, we get y = 3.
Putting x = 2, we get y = 6.
Putting x = −2, we get y = 0.
Thus, we have the following table for the equation $\frac{3}{2}x-y+3=0$.

x	0	2	−2
y	3	6	0

Now, plot the points P(0, 3), Q(2, 6) and R (−2, 0) on the same graph paper as above. Join PQ and PR to get the graph line QR. Extend it on both ways.
Then, QR is the graph of the equation $\frac{3}{2}x-y+3=0$ and it meets the y-axis at P(0, 3).

It is clear from the graph that the two graph lines are parallel and do not intersect even when produced.
Hence, the given system of equations has no solutions.