Graphical Representation of a Pair of Linear Equations

Video Explanation

Question

Represent the following pair of equations graphically and write the coordinates of the points where the lines intersect the y-axis:

\[ x + 3y = 6 \]

\[ 2x – 3y = 12 \]

Solution

Step 1: Write Both Equations in the Form \(y = mx + c\)

Equation (1):

\[ x + 3y = 6 \Rightarrow 3y = 6 – x \Rightarrow y = 2 – \frac{x}{3} \]

Equation (2):

\[ 2x – 3y = 12 \Rightarrow -3y = 12 – 2x \Rightarrow y = \frac{2}{3}x – 4 \]

Step 2: Prepare Tables of Values

For Equation (1): \(y = 2 – \frac{x}{3}\)

x	y
0	2
6	0

For Equation (2): \(y = \frac{2}{3}x – 4\)

x	y
0	-4
6	0

Step 3: Graphical Representation

Plot the following points on the same Cartesian plane:

• Line 1: (0, 2) and (6, 0)
• Line 2: (0, −4) and (6, 0)

Join each pair of points to obtain two straight lines.

Points of Intersection with the Y-Axis

For the line \(x + 3y = 6\), when \(x = 0\):

\[ y = 2 \Rightarrow \text{Point} = (0, 2) \]

For the line \(2x – 3y = 12\), when \(x = 0\):

\[ y = -4 \Rightarrow \text{Point} = (0, -4) \]

Answer

The coordinates of the points where the given lines intersect the y-axis are:

• (0, 2)
• (0, −4)

Conclusion

The graphs of the given equations are straight lines, and their points of intersection with the y-axis are (0, 2) and (0, −4).