Graph of Linear Equations and Area of the Quadrilateral

Video Explanation

Question

Draw the graph of the equations:

\[ x = 3 \]

\[ x = 5 \]

\[ 2x – y – 4 = 0 \]

Also find the area of the quadrilateral formed by these lines and the x-axis.

Solution

Step 1: Write the Equation in Convenient Form

Given:

\[ 2x – y – 4 = 0 \Rightarrow y = 2x – 4 \]

Step 2: Points of Intersection

Intersection with x-axis \((y = 0)\)

\[ 0 = 2x – 4 \Rightarrow x = 2 \]

So the line meets x-axis at (2, 0).

Intersection with \(x = 3\)

\[ y = 2(3) – 4 = 2 \Rightarrow (3, 2) \]

Intersection with \(x = 5\)

\[ y = 2(5) – 4 = 6 \Rightarrow (5, 6) \]

Step 3: Graphical Representation

Plot the lines:

• \(x = 3\) (vertical line)
• \(x = 5\) (vertical line)
• \(y = 2x – 4\)
• x-axis \((y = 0)\)

These lines form a quadrilateral.

Step 4: Vertices of the Quadrilateral

The vertices are:

• \((3, 0)\)
• \((5, 0)\)
• \((5, 6)\)
• \((3, 2)\)

Step 5: Area of the Quadrilateral

The quadrilateral is a trapezium with parallel sides along the y-direction.

Height at \(x = 3\) = 2 units

Height at \(x = 5\) = 6 units

Distance between parallel sides = \(5 – 3 = 2\) units

\[ \text{Area} = \frac{1}{2} \times (2 + 6) \times 2 = 8 \]

Answer

Area of the quadrilateral formed by the given lines and the x-axis is:

8 square units

Conclusion

By drawing the graphs of the given equations, the required quadrilateral is obtained and its area is 8 square units.