Consistent Pair of Linear Equations Explained

 

❓ Question

Check graphically whether the pair of linear equations:

$$x+3y=6$$

and

$$2x-3y=12$$

is consistent.
If yes, solve them graphically.

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🖼️ Solution Image

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✍️ Short Concept

A pair of linear equations is:

• Consistent if the lines intersect or coincide.
• Inconsistent if the lines are parallel.

For graphical solution:

✅ Convert equations into points
✅ Plot both lines
✅ Intersection point gives the solution

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🔷 Step 1 — Equation 1

Given:

$$x+3y=6$$

Take:

When $x=0$

$$3y=6$$
$$y=2$$

Point:

$$(0,2)$$

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When $y=0$

$$x=6$$

Point:

$$(6,0)$$

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🔷 Step 2 — Equation 2

Given:

$$2x-3y=12$$

Take:

When $x=0$

$$-3y=12$$
$$y=-4$$

Point:

$$(0,-4)$$

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When $y=0$

$$2x=12$$
$$x=6$$

Point:

$$(6,0)$$

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🔷 Step 3 — Graphical Observation

After plotting both equations:

• First line passes through:

$$(0,2),\ (6,0)$$

• Second line passes through:

$$(0,-4),\ (6,0)$$

Both lines intersect at:

$$\boxed{(6,0)}$$

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🔷 Step 4 — Conclusion

Since the two lines intersect at one point,

✅ The pair of equations is consistent.

And the graphical solution is:

$$\boxed{x=6,\ y=0}$$

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✅ Final Answer

$$\boxed{(6,0)}$$

The pair of equations is:

$$\boxed{\text{Consistent}}$$

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⭐ Important Concept

For two linear equations:

• Intersecting lines → One solution → Consistent
• Parallel lines → No solution → Inconsistent
• Coincident lines → Infinite solutions → Consistent

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📚 Related Topics

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