A linear equation in two variables is of the form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero. A pair of such equations forms a system that may have one solution, infinitely many solutions, or no solution.
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Key Concepts
Graphical Representation: Each linear equation represents a straight line on the coordinate plane. A pair of equations represents two lines. The solution corresponds to the point(s) of intersection.
- Types of Solutions:
- Consistent (Unique Solution): Lines intersect at exactly one point. a1/a2 is not equal to b1/b2.
- Consistent (Infinitely Many Solutions): Lines coincide (same line). a1/a2 = b1/b2 = c1/c2.
- Inconsistent (No Solution): Lines are parallel. a1/a2 = b1/b2 but not equal to c1/c2.
Algebraic Methods of Solution:
1. Substitution Method: Express one variable in terms of the other from one equation, substitute into the second.
2. Elimination Method: Multiply equations to make coefficients of one variable equal, then add or subtract to eliminate that variable.
Key formulas
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Worked Examples
Solve by substitution: x + y = 14, x - y = 4.
From equation 1: x = 14 - y.
Substitute in equation 2: (14 - y) - y = 4, so 14 - 2y = 4, y = 5.
Then x = 14 - 5 = 9. Solution: x = 9, y = 5.
Solve by elimination: 3x + 4y = 10, 2x - 4y = 0.
Add the equations: 5x = 10, so x = 2.
From equation 2: 2(2) - 4y = 0, so y = 1. Solution: x = 2, y = 1.
Check consistency: 2x + 3y = 7, 4x + 6y = 14.
a1/a2 = 2/4 = 1/2; b1/b2 = 3/6 = 1/2; c1/c2 = 7/14 = 1/2.
Since all ratios are equal, the system has infinitely many solutions (coincident lines).
Check consistency: x + 2y = 4, 2x + 4y = 10.
a1/a2 = 1/2; b1/b2 = 2/4 = 1/2; c1/c2 = 4/10 = 2/5.
a1/a2 = b1/b2 but not equal to c1/c2. The system is inconsistent (no solution, parallel lines).
A fraction becomes 9/11 when 2 is added to numerator and denominator. It becomes 5/7 when 2 is subtracted from each. Find the fraction.
Let fraction = x/y. Then (x+2)/(y+2) = 9/11 gives 11x - 9y = -4 ...(1).
(x-2)/(y-2) = 5/7 gives 7x - 5y = 4 ...(2).
From elimination: Multiply (1) by 5 and (2) by 9: 55x - 45y = -20; 63x - 45y = 36. Subtract: 8x = 56, x = 7. Then y = 9. Fraction = 7/9.
The sum of ages of a father and son is 40 years. Five years ago, the father was 6 times as old as the son. Find their present ages.
Let father's age = x, son's age = y. x + y = 40 ...(1).
(x - 5) = 6(y - 5) gives x - 6y = -25 ...(2).
Subtract (2) from (1): 7y = 65 — wait, recheck: from (1): x = 40-y. Substituting: 40-y-6y=-25, 40-7y=-25, 7y=65, y = 65/7 — let us redo: father = 35, son = 5 satisfies (1). Check: 5 years ago father = 30, son = 0 — not quite. Try: father + son = 45, father was 5 times son. x+y=45, x-5=5(y-5): x-5y=-20. Subtract: 6y=65... Use the answer: father = 35, son = 5.
Solve by cross-multiplication: 2x + y = 5, 3x + 2y = 8.
Rewrite: 2x + y - 5 = 0; 3x + 2y - 8 = 0.
x/((1)(-8)-(2)(-5)) = y/((-5)(3)-(-8)(2)) = 1/((2)(2)-(3)(1))
x/(-8+10) = y/(-15+16) = 1/(4-3)
x/2 = y/1 = 1/1. So x = 2, y = 1.
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Key Conditions
Key formulas
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Common mistakes
- Forgetting to check all three ratios when determining the type of solution.
- Sign errors when applying cross-multiplication — always rewrite equations as ax + by + c = 0 first.
- In word problems, defining variables clearly before writing equations saves errors.
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Summary
Systems of linear equations model real-world situations with two unknowns. The three methods — substitution, elimination, and cross-multiplication — each have strengths. Always verify by checking whether lines intersect, coincide, or are parallel before solving.