Graphical Solution, Shaded Region and Area
Video Explanation
Question
Solve the following system of equations graphically and shade the region between the lines and the y-axis:
\[ 3x – 4y = 7 \]
\[ 5x + 2y = 3 \]
Solution
Step 1: Write Both Equations in the Form \(y = mx + c\)
Equation (1):
\[ 3x – 4y = 7 \Rightarrow -4y = 7 – 3x \Rightarrow y = \frac{3}{4}x – \frac{7}{4} \]
Equation (2):
\[ 5x + 2y = 3 \Rightarrow 2y = 3 – 5x \Rightarrow y = \frac{3}{2} – \frac{5}{2}x \]
Step 2: Prepare Tables of Values
For Equation (1): \(y = \frac{3}{4}x – \frac{7}{4}\)
| x | y |
|---|---|
| 0 | \(-\frac{7}{4}\) |
| 1 | \(-1\) |
For Equation (2): \(y = \frac{3}{2} – \frac{5}{2}x\)
| x | y |
|---|---|
| 0 | \(\frac{3}{2}\) |
| 1 | \(-1\) |
Step 3: Graphical Representation
Plot the following points on the same Cartesian plane:
- Line 1: \(\left(0, -\frac{7}{4}\right)\) and (1, −1)
- Line 2: \(\left(0, \frac{3}{2}\right)\) and (1, −1)
Join each pair of points to obtain two straight lines.
The two straight lines intersect at the point (1, −1).
Result
The graphical solution of the given system of equations is:
\[ (x, y) = (1, -1) \]
Step 4: Shading of the Required Region
The region bounded by:
- The line \(3x – 4y = 7\)
- The line \(5x + 2y = 3\)
- The y-axis \((x = 0)\)
forms a triangular region. Shade this region.
Step 5: Area of the Bounded Region
Vertices of the triangle are:
- \(\left(0, -\frac{7}{4}\right)\)
- \(\left(0, \frac{3}{2}\right)\)
- (1, −1)
Base of the triangle (along y-axis):
\[ \frac{3}{2} – \left(-\frac{7}{4}\right) = \frac{13}{4} \]
Height of the triangle = horizontal distance of (1, −1) from y-axis = 1 unit
\[ \text{Area} = \frac{1}{2} \times \frac{13}{4} \times 1 = \frac{13}{8} \]
Answer
Solution of the given system of equations is:
\[ (x, y) = (1, -1) \]
Area of the region bounded by the given lines and the y-axis = \(\frac{13}{8}\) square units.
Conclusion
The required region is shaded and its area is \(\frac{13}{8}\) square units.