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📑 In This Chapter Guide (Table of Contents)
1. Standard Form & Consistency Conditions
A pair of linear equations in variables x and y is represented as:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
| Ratio Comparison | Graphical Representation | Algebraic Interpretation | System Consistency |
|---|---|---|---|
a₁/a₂ ≠ b₁/b₂ |
Intersecting lines at 1 point | Exactly 1 Unique Solution | Consistent |
a₁/a₂ = b₁/b₂ = c₁/c₂ |
Coincident lines (overlapping) | Infinitely Many Solutions | Consistent (Dependent) |
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
Parallel lines (never meet) | No Solution (Zero) | Inconsistent |
2. Algebraic Solution Methods: Substitution & Elimination
Method 1: Elimination Method (Most Efficient for Board Exams):
- Multiply one or both equations by suitable non-zero constants so coefficients of either x or y become equal.
- Add or subtract the two equations to eliminate that variable.
- Solve the resulting linear equation in one variable.
- Substitute this value back into either original equation to find the other variable.
Method 2: Substitution Method: Express one variable in terms of the other from one equation, and substitute it into the second equation.
3. Master Formula for Upstream & Downstream Boat Word Problems
Let speed of boat in still water = x km/h, and speed of stream (current) = y km/h:
- Speed Downstream (Moving with the stream):
Speed = (x + y) km/h - Speed Upstream (Moving against the stream):
Speed = (x - y) km/h(Always x > y) - Time Equations:
Time = Distance / Speed.
Upstream Time:t₁ = Distance / (x - y); Downstream Time:t₂ = Distance / (x + y).
💡 Frequently Asked Questions (FAQ)
❓ For what value of k will the equations 2x + 3y = 7 and (k-1)x + (k+2)y = 3k have infinitely many solutions?
For infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂. Thus: 2/(k-1) = 3/(k+2) = 7/(3k). Cross-multiplying the first two: 2(k+2) = 3(k-1) ⟹ 2k + 4 = 3k - 3 ⟹ k = 7. Checking with 7/(3k): 7/(21) = 1/3, which matches 2/(7-1) = 2/6 = 1/3. Hence, k = 7.
❓ Solve for x and y: 2x + 3y = 11 and 2x - 4y = -24.
Subtracting equation 2 from equation 1 eliminates x: (2x - 2x) + (3y - (-4y)) = 11 - (-24) ⟹ 7y = 35 ⟹ y = 5. Substituting y = 5 into 2x + 3(5) = 11 ⟹ 2x = -4 ⟹ x = -2. Solution: x = -2, y = 5.
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