Forming Linear Equations for Different Graphical Representations

 

❓ Question

Given the linear equation

$$2x+3y-8=0$$

write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) Intersecting lines

(ii) Parallel lines

(iii) Coincident lines

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🖼️ Solution Image

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✍️ Short Concept

This question is based on:

👉 Pair of Linear Equations in Two Variables

👉 Conditions for Intersecting, Parallel and Coincident Lines

👉 Comparing Ratios of Coefficients

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🔷 Step 1 — Recall the Conditions

For two equations

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

Intersecting Lines

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

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Parallel Lines

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

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Coincident Lines

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

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🔷 Step 2 — Intersecting Lines

Given equation:

$$2x+3y-8=0$$

Choose another equation:

$$3x+2y-7=0$$

Checking:

$$\frac{2}{3}\ne\frac{3}{2}$$

Hence, the two lines intersect.

✅ Answer

$$\boxed{3x+2y-7=0}$$

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🔷 Step 3 — Parallel Lines

Given equation:

$$2x+3y-8=0$$

Choose:

$$4x+6y-12=0$$

Checking:

$$\frac{2}{4}=\frac{3}{6}=\frac12$$

but

$$\frac{-8}{-12}=\frac23$$

Thus,

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

Hence, lines are parallel.

✅ Answer

$$\boxed{4x+6y-12=0}$$

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🔷 Step 4 — Coincident Lines

Given equation:

$$2x+3y-8=0$$

Multiply by 2:

$$4x+6y-16=0$$

Checking:

$$\frac{2}{4} = \frac{3}{6} = \frac{-8}{-16} = \frac12$$

Hence, both equations represent the same line.

✅ Answer

$$\boxed{4x+6y-16=0}$$

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📋 Final Answers

Case	Another Equation
(i) Intersecting Lines	$\boxed{3x+2y-7=0}$
(ii) Parallel Lines	$\boxed{4x+6y-12=0}$
(iii) Coincident Lines	$\boxed{4x+6y-16=0}$

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🚨 Common Mistakes

❌ Using equal coefficient ratios for intersecting lines

❌ Making all three ratios equal while trying to form parallel lines

❌ Forgetting that coincident lines are simply multiples of the same equation

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✅ Final Takeaway

For a pair of linear equations:

✔ Intersecting lines → Unique solution

✔ Parallel lines → No solution

✔ Coincident lines → Infinitely many solutions

These can be identified by comparing:

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

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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