CBSE · Class 10 · Mathematics · Chapter 3

Pair of Linear Equations in Two Variables — Formula Sheet

Board Formulas

11 formulas

1.General Form of a Pair
$a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$
$x, y$ : The two unknowns (variables) · $a_{1}, b_{1}, a_{2}, b_{2}$ : Coefficients of x and y in the two equations · $c_{1}, c_{2}$ : Constant terms
a₁, b₁, c₁, a₂, b₂, c₂ are real numbers with a₁² + b₁² ≠ 0 and a₂² + b₂² ≠ 0. The graph of each equation is a straight line.
2.Axis Intercepts of a Line
$a x + b y + c = 0 : x-intercept = - \frac{c}{a}, y-intercept = - \frac{c}{b}$
Quickest two points to plot: (−c/a, 0) on the x-axis and (0, −c/b) on the y-axis. Put y = 0, then x = 0 (a, b ≠ 0).
3.Solution of the Pair
$(x_{0}, y_{0}) is a solution ⟺ a_{1} x_{0} + b_{1} y_{0} + c_{1} = 0 and a_{2} x_{0} + b_{2} y_{0} + c_{2} = 0$
A solution must satisfy BOTH equations. Graphically it is a common point of the two lines.
4.Intersecting Lines — Unique Solution★
$\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}} \Rightarrow unique solution$
The lines intersect at exactly one point, so the pair is consistent. c₁/c₂ does not matter here.
5.Coincident Lines — Infinitely Many Solutions★
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}} \Rightarrow infinitely many solutions$
One equation is a multiple of the other, so both describe the same line and every point on it is a solution. The pair is dependent (and consistent).
6.Parallel Lines — No Solution★
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}} \Rightarrow no solution$
The lines are parallel and distinct, so they never meet: the pair is inconsistent. A pair is consistent if it has at least one solution (unique or infinitely many).
7.Substitution Method
$y = - \frac{a _{1} x + c _{1}}{b _{1}} (b_{1} \neq = 0) \to substitute into the second equation$
Express one variable in terms of the other from one equation, substitute into the other to get an equation in one variable, solve, then back-substitute.
8.Elimination Method★
$a_{2} (a_{1} x + b_{1} y + c_{1}) - a_{1} (a_{2} x + b_{2} y + c_{2}) = 0 \Rightarrow (a_{2} b_{1} - a_{1} b_{2}) y = a_{1} c_{2} - a_{2} c_{1}$
Multiply the equations so that one variable has equal coefficients, then subtract (or add if the signs are opposite) to remove it. In practice multiply by the smallest numbers that do the job.
9.When Both Variables Disappear
$0 = 0 \Rightarrow infinitely many solutions; 0 = k (k \neq = 0) \Rightarrow no solution$
In substitution or elimination, if both variables cancel, a true statement means coincident lines and a false statement means parallel lines.
10.Two-digit Number Set-up
$Number = 10 t + u, Reversed number = 10 u + t$
$t$ : Tens digit · $u$ : Units digit
t = tens digit, u = units digit. Note that (10u + t) − (10t + u) = 9(u − t).
11.Age Problem Set-up
$Age n years ago = x - n, Age n years hence = x + n$
Let x be the PRESENT age. Write each condition using the same present-age variables, then solve the pair.