Infinite Solutions in Pair of Linear Equations

 

❓ Concept Question

What happens when two linear equations represent the same line, and how can we identify whether the system has infinitely many solutions?

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🖼️ Concept Image

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✍️ Short Concept

This concept is based on:

👉 Pair of Linear Equations in Two Variables

👉 Dependent (Coincident) Lines

👉 Infinitely Many Solutions

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🔷 Step 1 — Form the Equations

Let:

• ₹$x$ = cost of one pencil
• ₹$y$ = cost of one eraser

From the question:

• Cost of 2 pencils and 3 erasers is ₹9

  $$2x+3y=9$$
  

• Cost of 4 pencils and 6 erasers is ₹18

  $$4x+6y=18$$
  

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🔷 Step 2 — Compare the Two Equations

Multiply the first equation by 2:

$$2(2x+3y)=2\times 9$$
$$4x+6y=18$$

This is exactly the same as the second equation.

So, both equations represent the same straight line.

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🔷 Step 3 — Check the Condition for Coincident Lines

For two linear equations:

$$a_1x+b_1y+c_1=0$$
$$a_2x+b_2y+c_2=0$$

If

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

then the two lines are coincident (overlap each other completely).

For this question:

$$2x+3y-9=0$$
$$4x+6y-18=0$$

So,

$$\frac{2}{4}=\frac{3}{6}=\frac{-9}{-18}=\frac{1}{2}$$

Hence, the lines are coincident.

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🔷 Step 4 — Number of Solutions

Since both equations represent the same line, every point on that line satisfies both equations.

Therefore, the pair of equations has:

✅ Infinitely many solutions.

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🔷 Step 5 — Can We Find the Cost of One Pencil and One Eraser?

No. A unique cost cannot be determined because there are infinitely many possible pairs $(x,y)$ that satisfy the equation.

For example:

• If $x=3$, then

  $$2(3)+3y=9$$
  $$y=1$$
  

• If $x=1.5$, then

  $$2(1.5)+3y=9$$
  $$y=2$$
  

Both pairs satisfy the given conditions.

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🚨 Common Mistakes

❌ Thinking that two equations always give one unique answer.

❌ Not noticing that one equation is simply a multiple of the other.

❌ Forgetting the condition:

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

for coincident lines.

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✅ Final Takeaway

If one linear equation is an exact multiple of the other, then both represent the same line.

This means:

✔ The lines are coincident.

✔ The pair of equations is consistent.

✔ The system has infinitely many solutions.

✔ A unique value of each variable cannot be determined.

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⭐ Class 10 Insight

For a pair of linear equations:

• $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$ → One unique solution.
• $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$→ No solution (parallel lines).
• $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ → Infinitely many solutions (coincident lines).

Remember this test—it is one of the most important concepts in this chapter.

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📚 Related Topics

• Find Zeroes and Verify Relationship Between Coefficients
• Find Quadratic Polynomial from Sum and Product of Zeroes
• Quadratic Polynomial with Irrational Zeroes Explained