Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables tests graphical meaning, consistency, algebraic solution methods, and word problems. A pair is consistent if it has at least one solution.
Core rules
Write equations as a1x+b1y+c1=0 and a2x+b2y+c2=0.
- If a1/a2 ≠ b1/b2, the lines intersect at one point: unique solution.
- If a1/a2 = b1/b2 = c1/c2, the lines coincide: infinitely many solutions.
- If a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel: no solution. Substitution and elimination both reduce the pair to one variable. Graphical solutions must be read from actual intersection points, not estimates.
Worked example
Solve x+y=10 and x-y=2. Adding gives 2x=12, so x=6. Substituting into x+y=10 gives y=4. Check in the second equation: 6-4=2. The pair is consistent and independent because the slopes are different; the solution is (6,4).
Traps
Compare ratios only after both equations are in standard form. A common error is to compare c1/c2 before checking a and b ratios. Parallel lines have no solution, while coincident lines have infinitely many. In word problems, define variables clearly and check whether negative or fractional answers make sense.