Question

$$\begin{gathered} 2x-3y=1, \\ 3x-4y=1. \end{gathered}$$

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 2x − 3y = 1

2x − 3y = 1
⇒ 3y = (2x − 1)
∴ $y=\frac{2x-1}{3}$ ...(i)
Putting x = −1, we get y = −1
Putting x = 2, we get y = 1
Putting x = 5, we get y = 3
Thus, we have the following table for the equation 2x − 3y = 1.

x	−1	2	5
y	−1	1	3

Now, plot the points A(−1, −1), B(2, 1) and C(5, 3) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 2x − 3y = 1.

Graph of 3x − 4y = 1
3x − 4y = 1
⇒ 4y = (3x − 1)
∴ $y=\frac{3x-1}{4}$ ...(ii)
Putting x = −1, we get y = −1
Putting x = 3, we get y = 2
Putting x = −5, we get y = −4
Thus, we have the following table for the equation 3x − 4y = 1.

x	−1	3	−5
y	−1	2	−4

Now, plot the points P(3, 2) and Q(−5, −4). The point A(−1, −1) has already been plotted. Join PA and QA and extend it on both ways.
Thus, PQ is the graph of 3x − 4y = 1.

The two graph lines intersect at A(−1 , −1).
∴ x = −1 and y = −1 are the solutions of the given system of equations.