Graph of Linear Equations and Area of the Triangle
Video Explanation
Question
Draw the graphs of the equations and determine the coordinates of the vertices of the triangle formed by these lines and the y-axis. Also find the area of the triangle:
\[ 5x – y = 5 \]
\[ 3x – y = 3 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 5x – y = 5 \Rightarrow y = 5x – 5 \]
Equation (2):
\[ 3x – y = 3 \Rightarrow y = 3x – 3 \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = 5x – 5\)
| x | y |
|---|---|
| 0 | -5 |
| 1 | 0 |
For Equation (2): \(y = 3x – 3\)
| x | y |
|---|---|
| 0 | -3 |
| 1 | 0 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, −5) and (1, 0)
- Line 2: (0, −3) and (1, 0)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (1, 0).
Step 4: Vertices of the Triangle with the Y-Axis
The triangle is formed by:
- Intersection of \(5x – y = 5\) with y-axis → (0, −5)
- Intersection of \(3x – y = 3\) with y-axis → (0, −3)
- Intersection point of the two lines → (1, 0)
Step 5: Area of the Triangle
Base of the triangle (along y-axis) = distance between (0, −5) and (0, −3) = 2 units
Height of the triangle = horizontal distance of point (1, 0) from y-axis = 1 unit
\[ \text{Area} = \frac{1}{2} \times 2 \times 1 = 1 \]
Answer
Vertices of the triangle are:
- (0, −5)
- (0, −3)
- (1, 0)
Area of the triangle = 1 square unit.
Conclusion
By drawing the graphs of the given equations, the triangle formed with the y-axis is obtained and its area is 1 square unit.