Graph of Linear Equations and Triangle Formed with the Y-Axis
Video Explanation
Question
Draw the graphs of the following equations on the same graph paper and find the coordinates of the vertices of the triangle formed by the two straight lines and the y-axis:
\[ 2x + 3y = 12 \]
\[ x – y = 1 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 2x + 3y = 12 \Rightarrow 3y = 12 – 2x \Rightarrow y = 4 – \frac{2}{3}x \]
Equation (2):
\[ x – y = 1 \Rightarrow y = x – 1 \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = 4 – \frac{2}{3}x\)
| x | y |
|---|---|
| 0 | 4 |
| 6 | 0 |
For Equation (2): \(y = x – 1\)
| x | y |
|---|---|
| 0 | -1 |
| 1 | 0 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, 4) and (6, 0)
- Line 2: (0, −1) and (1, 0)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (3, 2).
Step 4: Vertices of the Triangle Formed with the Y-Axis
The triangle is formed by:
- Intersection of line \(2x + 3y = 12\) with the y-axis → (0, 4)
- Intersection of line \(x – y = 1\) with the y-axis → (0, −1)
- Intersection point of the two lines → (3, 2)
Answer
The coordinates of the vertices of the triangle formed by the two straight lines and the y-axis are:
- \((0, 4)\)
- \((0, -1)\)
- \((3, 2)\)
Conclusion
The required triangle is obtained by drawing the graphs of the given equations and the y-axis on the same graph paper.