Infinite Solutions in Pair of Linear Equations Explained

 

❓ Question

Graphically, find whether the following pair of linear equations has:

• no solution
• unique solution
• infinitely many solutions

$$5x-8y+1=0$$

and

$$3x-\frac{24}{5}y+\frac{3}{5}=0$$

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🖼️ Solution Image

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✍️ Short Concept

For a pair of linear equations:

• Intersecting lines → Unique solution
• Parallel lines → No solution
• Coincident lines → Infinitely many solutions

To check graphically:

✅ Find points for each line
✅ Draw both lines
✅ Observe their position

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🔷 Step 1 — First Equation

Given:

$$5x-8y+1=0$$

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When $x=0$

$$-8y+1=0$$
$$y=\frac{1}{8}$$

Point:

$$\left(0,\frac18\right)$$

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When $y=0$

$$5x+1=0$$
$$x=-\frac15$$

Point:

$$\left(-\frac15,0\right)$$

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🔷 Step 2 — Second Equation

Given:

$$3x-\frac{24}{5}y+\frac35=0$$

Multiply by 5:

$$15x-24y+3=0$$

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When $x=0$

$$-24y+3=0$$
$$y=\frac18$$

Point:

$$\left(0,\frac18\right)$$

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When $y=0$

$$15x+3=0$$
$$x=-\frac15$$

Point:

$$\left(-\frac15,0\right)$$

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🔷 Step 3 — Graphical Observation

Both equations pass through the same points:

$$\left(0,\frac18\right)$$

and

$$\left(-\frac15,0\right)$$

So both lines lie exactly on each other.

Hence, the two lines are:

$$\boxed{\text{Coincident Lines}}$$

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🔷 Step 4 — Conclusion

Since the two lines are coincident,

✅ The pair of equations has:

$$\boxed{\text{Infinitely many solutions}}$$

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✅ Final Answer

$$\boxed{\text{The system has infinitely many solutions.}}$$

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⭐ Important Concept

For equations:

$$a_1x+b_1y+c_1=0$$

and

$$a_2x+b_2y+c_2=0$$

If:

$$\frac{a_1}{a_2}= \frac{b_1}{b_2}= \frac{c_1}{c_2}$$

then lines are coincident and solutions are infinite.

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📚 Related Topics

• Zeroes and Coefficients of Quadratic Polynomial
• Number of Zeroes of a Polynomial from Graphs
• Prove that 3 + 2√5 is Irrational