Graphical Solution and Area of Triangles with the X-Axis

Video Explanation

Question

Draw the graphs of the pair of linear equations:

\[ x – y + 2 = 0 \]

\[ 4x – y – 4 = 0 \]

Calculate the area of the triangles formed by the lines so drawn and the x-axis.

Solution

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

Equation (1):

\[ x – y + 2 = 0 \Rightarrow y = x + 2 \]

Equation (2):

\[ 4x – y – 4 = 0 \Rightarrow y = 4x – 4 \]

Step 2: Prepare Tables of Values

For Equation (1): \(y = x + 2\)

x	y
0	2
-2	0

For Equation (2): \(y = 4x – 4\)

x	y
0	-4
1	0

Step 3: Graphical Representation

Plot the above points on the same Cartesian plane and draw the two straight lines.

The two lines intersect at the point:

\[ x + 2 = 4x – 4 \Rightarrow 3x = 6 \Rightarrow x = 2,\; y = 4 \]

Intersection point = \((2, 4)\)

Triangle Formed with the X-Axis by Line \(y = x + 2\)

Intercept on x-axis: \((-2, 0)\)

Intercept on y-axis: \((0, 2)\)

Base = 2 units, Height = 2 units

\[ \text{Area}_1 = \frac{1}{2} \times 2 \times 2 = 2 \]

Triangle Formed with the X-Axis by Line \(y = 4x – 4\)

Intercept on x-axis: \((1, 0)\)

Intercept on y-axis: \((0, -4)\)

Base = 1 unit, Height = 4 units

\[ \text{Area}_2 = \frac{1}{2} \times 1 \times 4 = 2 \]

Answer

Area of triangle formed by \(x – y + 2 = 0\) and the x-axis = 2 square units

Area of triangle formed by \(4x – y – 4 = 0\) and the x-axis = 2 square units

Conclusion

Thus, the areas of the triangles formed by each line with the x-axis are equal and each is 2 square units.