Ratio Comparison Method for Linear Equations

 

❓ Question

On comparing the ratios:

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

find out whether the following pairs of linear equations are:

✔ Consistent
✔ or Inconsistent

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(i)

$$3x+2y=5$$
$$2x-3y=7$$

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🖼️ Solution Image

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✍️ Short Concept

For equations:

$$a_1x+b_1y+c_1=0$$

and

$$a_2x+b_2y+c_2=0$$

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🔷 Step 1 — Conditions for Consistency

Case 1 — Unique Solution

If:

$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$$

then lines intersect at one point.

✔ Pair is consistent

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Case 2 — Infinite Solutions

If:

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

then lines are coincident.

✔ Pair is consistent

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Case 3 — No Solution

If:

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

then lines are parallel.

❌ Pair is inconsistent

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🔷 Step 2 — Solve Each Pair

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(i)

$$3x+2y=5$$
$$2x-3y=7$$

Comparing coefficients:

$$\frac{a_1}{a_2}=\frac32$$
$$\frac{b_1}{b_2}=\frac2{-3}$$

Since:

$$\frac32\ne\frac2{-3}$$

the lines intersect at one point.

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✅ Final Answer (i)

$$\boxed{\text{Consistent pair with unique solution}}$$

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(ii)

$$2x-3y=8$$
$$4x-6y=9$$

Comparing ratios:

$$\frac{a_1}{a_2}=\frac24=\frac12$$
$$\frac{b_1}{b_2}=\frac{-3}{-6}=\frac12$$
$$\frac{c_1}{c_2}=\frac89$$

Since:

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

the lines are parallel.

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✅ Final Answer (ii)

$$\boxed{\text{Inconsistent pair (No solution)}}$$

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(iii)

$$\frac{3x}{2}+\frac{5y}{3}=7$$
$$9x-10y=14$$

Comparing ratios:

$$\frac{a_1}{a_2}=\frac{3/2}{9}=\frac16$$
$$\frac{b_1}{b_2}=\frac{5/3}{-10}=-\frac16$$

Since:

$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$$

the pair has a unique solution.

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✅ Final Answer (iii)

$$\boxed{\text{Consistent pair with unique solution}}$$

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(iv)

$$5x-3y=11$$
$$-10x+6y=-22$$

Comparing ratios:

$$\frac{a_1}{a_2}=\frac5{-10}=-\frac12$$
$$\frac{b_1}{b_2}=\frac{-3}{6}=-\frac12$$
$$\frac{c_1}{c_2}=\frac{11}{-22}=-\frac12$$

All ratios are equal.

Hence lines are coincident.

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✅ Final Answer (iv)

$$\boxed{\text{Consistent pair with infinitely many solutions}}$$

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(v)

$$\frac43x+2y=8$$
$$2x+3y=12$$

Comparing ratios:

$$\frac{a_1}{a_2}=\frac{4/3}{2}=\frac23$$
$$\frac{b_1}{b_2}=\frac23$$
$$\frac{c_1}{c_2}=\frac8{12}=\frac23$$

All ratios are equal.

Hence lines are coincident.

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✅ Final Answer (v)

$$\boxed{\text{Consistent pair with infinitely many solutions}}$$

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⭐ Important Concept

For pair of linear equations:

✔ Intersecting lines → Unique solution
✔ Coincident lines → Infinite solutions
✔ Parallel lines → No solution

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🚨 Common Mistakes

❌ Forgetting to convert equations into standard form

❌ Comparing wrong coefficients

❌ Writing consistent instead of inconsistent for parallel lines

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✅ Final Takeaway

Use ratio comparison:

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

to quickly determine whether equations are:

• intersecting
• parallel
• or coincident.

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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