Two Lines Only Ever Do One of Three Things Pair of Linear Equations in Two Variables

At a school fair, Akhila rides the Giant Wheel (Rs.3 per ride) and plays Hoopla (Rs.4 per go), spending Rs.20 in total, with her number of Hoopla goes being exactly half her number of rides. Calling the rides x and the Hoopla goes y gives two equations at once: y = x/2, and 3x + 4y = 20. This is a pair of linear equations in two variables, and finding values of x and y that satisfy BOTH equations simultaneously is the whole problem this chapter solves.

Every pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represents two straight lines, and those two lines can only ever relate in exactly three ways. If a1/a2 is not equal to b1/b2, the lines intersect at exactly one point, giving one unique solution (called consistent). If a1/a2 equals b1/b2 but not c1/c2, the lines are parallel and never meet, giving no solution at all (called inconsistent). If all three ratios, a1/a2, b1/b2 and c1/c2, are equal, the two equations actually describe the exact same line, coinciding completely and giving infinitely many shared solutions (called consistent and dependent). Just comparing these three ratios, without drawing a single point, reveals which of the three situations is in play.

The substitution method solves one equation for one variable in terms of the other, then substitutes that expression into the second equation, collapsing two unknowns down to one. Solving Aftab's age riddle this way ('seven years ago I was seven times as old as you; three years from now I'll be three times as old') gives Aftab's present age as 42 and his daughter's as 12. But substitution can also reveal the two edge cases directly: sometimes the substitution collapses to a true statement like 18 = 18 for every value (meaning infinitely many solutions exist, the equations are secretly the same line), and sometimes it collapses to a false statement like -4 = 0 (meaning no solution exists at all, the lines are parallel and never meet, exactly like two railway tracks that never cross).

The elimination method instead scales one or both equations so that one variable's coefficients become identical (or exact opposites), then adds or subtracts the equations to make that variable disappear entirely. Two friends whose incomes are in ratio 9:7, and whose expenditures are in ratio 4:3, both saving exactly Rs.2000 a month, can have their actual incomes recovered this way: scaling the income and expenditure relationships and eliminating one variable reveals incomes of Rs.18,000 and Rs.14,000. As with substitution, elimination can also collapse to a false numerical statement, immediately signalling that no solution exists for that particular pair of equations.

A classic word-problem type asks for a 2-digit number where the number plus its own digit-reversal equals 66, and the digits differ by 2. Calling the tens digit x and the units digit y, the original number is 10x + y and its reversal is 10y + x, giving the equation (10x+y) + (10y+x) = 66, which simplifies to x + y = 6. Combined with the digit-difference condition x - y = 2 (or y - x = 2), solving the pair gives two genuinely valid answers, 42 and 24, since the digit-difference condition could point in either direction without further information ruling one out.