Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions: 5x – 8y + 1 = 0 (1) 3x – 24 5 y + 3 5 = 0 (2)
Step-by-Step Solution
Key Concept: For a pair of linear equations in two variables, write each equation in the slope‑intercept form $y=mx+c$. Compare the slopes ($m$) and intercepts ($c$):<br>- If slopes are different, the lines intersect at a unique point → unique solution.<br>- If slopes are equal but intercepts are different, the lines are parallel → no solution.<br>- If both slopes and intercepts are equal, the lines coincide → infinitely many solutions.
1. Convert each equation to slope‑intercept form
- Equation (1): $5x-8y+1=0 \Rightarrow -8y = -5x-1 \Rightarrow y = \frac{5}{8}x + \frac{1}{8}$
Hence, slope $m_1 = \frac{5}{8}$ and $y$‑intercept $c_1 = \frac{1}{8}$.
- Equation (2): $3x-\frac{24}{5}y+\frac{3}{5}=0$.
Multiply by $5$ to clear fractions: $15x-24y+3=0 \Rightarrow -24y = -15x-3 \Rightarrow y = \frac{15}{24}x + \frac{3}{24}$.
Simplify: $\frac{15}{24}=\frac{5}{8}$ and $\frac{3}{24}=\frac{1}{8}$.
Thus, $y = \frac{5}{8}x + \frac{1}{8}$, giving slope $m_2 = \frac{5}{8}$ and intercept $c_2 = \frac{1}{8}$.
2. Compare slopes and intercepts
- $m_1 = m_2 = \frac{5}{8}$
- $c_1 = c_2 = \frac{1}{8}$
Since both the slopes and the intercepts are identical, the two equations represent the *same straight line*.
3. Conclusion (graphical interpretation)
- The two lines coincide; therefore they intersect at every point on the line.
- Hence, the system has infinitely many solutions.
4. Graphical description (for the figure)
- The graph would show a single straight line passing through the point $(0,\frac{1}{8})$ with a gentle positive slope $\frac{5}{8}$. Both equations plot the same line, so only one line appears on the coordinate plane.
Correct Answer: Infinitely many solutions (the two equations represent the same straight line).