Graphical Solution and Triangle Formed with the Y-Axis
Video Explanation
Question
Solve the following system of linear equations graphically and determine the coordinates of the vertices of the triangle formed by these lines and the y-axis:
\[ 4x – 5y – 20 = 0 \]
\[ 3x + 5y – 15 = 0 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 4x – 5y – 20 = 0 \Rightarrow -5y = -4x + 20 \Rightarrow y = \frac{4}{5}x – 4 \]
Equation (2):
\[ 3x + 5y – 15 = 0 \Rightarrow 5y = 15 – 3x \Rightarrow y = 3 – \frac{3}{5}x \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = \frac{4}{5}x – 4\)
| x | y |
|---|---|
| 0 | -4 |
| 5 | 0 |
For Equation (2): \(y = 3 – \frac{3}{5}x\)
| x | y |
|---|---|
| 0 | 3 |
| 5 | 0 |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: (0, −4) and (5, 0)
- Line 2: (0, 3) and (5, 0)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (5, 0).
Step 4: Vertices of the Triangle with the Y-Axis
The triangle is formed by:
- Intersection of line \(4x – 5y – 20 = 0\) with y-axis → (0, −4)
- Intersection of line \(3x + 5y – 15 = 0\) with y-axis → (0, 3)
- Intersection point of the two lines → (5, 0)
Answer
The coordinates of the vertices of the triangle formed by the given lines and the y-axis are:
- (0, −4)
- (0, 3)
- (5, 0)
Conclusion
By drawing the graphs of the given equations on the same graph paper, the required triangle with the y-axis is obtained.