Graphical Solution of Linear Equations in Two Variables

 

❓ Question

Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically.

(i)

$$x+y=5$$
$$2x+2y=10$$

(ii)

$$x-y=8$$
$$3x-3y=16$$

(iii)

$$2x+y-6=0$$
$$4x-2y-4=0$$

(iv)

$$2x-2y-2=0$$
$$4x-4y-5=0$$

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🖼️ Solution Image

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✍️ Short Concept

This question is based on:

👉 Pair of Linear Equations

👉 Consistent and Inconsistent Equations

👉 Graphical Interpretation of Lines

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🔷 Step 1 — Conditions for Consistency

For equations:

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

Case 1: Unique Solution

If

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

✔ Lines intersect

✔ Consistent pair

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Case 2: Infinite Solutions

If

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

✔ Coincident lines

✔ Consistent pair

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Case 3: No Solution

If

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

✔ Parallel lines

❌ Inconsistent pair

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🔷 Step 2 — Part (i)

$$x+y=5$$
$$2x+2y=10$$

Comparing coefficients:

$$\frac{a_1}{a_2}=\frac12$$
$$\frac{b_1}{b_2}=\frac12$$
$$\frac{c_1}{c_2}=\frac{-5}{-10}=\frac12$$

All ratios are equal.

Hence the lines are coincident.

✅ Answer

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

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🔷 Step 3 — Part (ii)

$$x-y=8$$
$$3x-3y=16$$

Comparing coefficients:

$$\frac{a_1}{a_2}=\frac13$$
$$\frac{b_1}{b_2}=\frac13$$
$$\frac{c_1}{c_2}=\frac{-8}{-16}=\frac12$$
$$\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$$

Lines are parallel.

✅ Answer

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

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🔷 Step 4 — Part (iii)

$$2x+y-6=0$$
$$4x-2y-4=0$$

Comparing coefficients:

$$\frac{a_1}{a_2}=\frac24=\frac12$$
$$\frac{b_1}{b_2}=\frac1{-2}=-\frac12$$

Since

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

the lines intersect at one point.

So the pair is consistent.

Finding the Solution

From

$$2x+y=6$$
$$y=6-2x$$

Substitute into

$$4x-2y=4$$
$$4x-2(6-2x)=4$$
$$4x-12+4x=4$$
$$8x=16$$
$$x=2$$

Putting in first equation:

$$y=6-2(2)=2$$

✅ Answer

$$\boxed{(x,y)=(2,2)}$$
$$\boxed{\text{Consistent pair with unique solution}}$$

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🔷 Step 5 — Part (iv)

$$2x-2y-2=0$$
$$4x-4y-5=0$$

Comparing coefficients:

$$\frac{a_1}{a_2}=\frac24=\frac12$$
$$\frac{b_1}{b_2}=\frac{-2}{-4}=\frac12$$
$$\frac{c_1}{c_2}=\frac{-2}{-5}$$

Since

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

the lines are parallel.

✅ Answer

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

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📋 Final Answers

Part	Result
(i)	Consistent, infinitely many solutions
(ii)	Inconsistent, no solution
(iii)	Consistent, unique solution $(2,2)$
(iv)	Inconsistent, no solution

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🚨 Common Mistakes

❌ Using only one ratio instead of comparing all three

❌ Forgetting to convert equations into standard form

❌ Confusing coincident lines with parallel lines

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✅ Final Takeaway

To determine consistency:

✔ Compare

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

✔ Intersecting lines → Unique solution

✔ Coincident lines → Infinite solutions

✔ Parallel lines → No solution

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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