Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables tests graphical and algebraic solutions and consistency. The exam expects you to identify unique, infinite or no solution from coefficients and to solve by substitution or elimination.
Core rules
For a1x+b1y+c1=0 and a2x+b2y+c2=0:
- a1/a2 ≠ b1/b2: unique solution, intersecting lines.
- a1/a2 = b1/b2 ≠ c1/c2: no solution, parallel lines.
- a1/a2 = b1/b2 = c1/c2: infinitely many solutions, coincident lines. Graphically, a solution is an intersection point. Algebraically, elimination removes one variable.
Worked example
Solve 2x+3y=12 and 3x-2y=5. Multiply first by 2: 4x+6y=24. Multiply second by 3: 9x-6y=15. Add: 13x=39, so x=3. Substitute in first: 6+3y=12, so y=2. Check second: 9-4=5. Solution (3,2).
Traps
Compare all three ratios before declaring consistency; equal a and b ratios with unequal c ratios means parallel, not unique. Keep signs when writing equations in standard form. A pair can look different after multiplying an equation-test ratios, not appearance. In word problems, define variables and check whether a negative or fractional answer is meaningful. Do not add equations without aligning like terms.