Maharashtra State Board · Class 10 · Mathematics · Algebra Chapter 1

Linear Equations in Two Variables — Formula Sheet

Board Formulas

16 formulas

1.General Form of a Linear Equation in Two Variables
$a x + b y + c = 0$
$x, y$ : Variables (unknowns) · $a, b$ : Coefficients of x and y (not both zero) · $c$ : Constant term
a, b, c are real numbers and a, b are not both zero. Its graph is a straight line; every point on the line is a solution of the equation.
2.Simultaneous Equations in Standard Form
$a_{1} x + b_{1} y = c_{1}, a_{2} x + b_{2} y = c_{2}$
Two equations in the same two variables. Bring both to this form (constants on the right) before using elimination or Cramer's rule.
3.Points for Drawing the Graph
$x = 0 \Rightarrow y = \frac{c}{b}, y = 0 \Rightarrow x = \frac{c}{a}$
For ax + by = c (a, b ≠ 0), putting x = 0 gives the point on the Y-axis and putting y = 0 gives the point on the X-axis. Plot at least four points in all, as the textbook advises.
4.Graphical Solution★
$Solution = (x, y) = point of intersection of the two lines$
Draw both lines on the same axes. The coordinates of the point where they meet satisfy both equations.
5.Substitution Method
$y = \frac{c _{1} - a _{1} x}{b _{1}}, b_{1} \neq = 0$
Express one variable in terms of the other from one equation and substitute it in the second equation. You get an equation in one variable.
6.Elimination Method (eliminating y)
$(a_{1} b_{2} - a_{2} b_{1}) x = c_{1} b_{2} - c_{2} b_{1}$
In practice, multiply the equations so that the coefficients of one variable become equal, then add or subtract to remove it.
7.Interchanged Coefficients (Add and Subtract)
$a x + b y = c, b x + a y = d \Rightarrow x + y = \frac{c + d}{a + b}, x - y = \frac{c - d}{a - b}$
Adding the equations gives (a + b)(x + y) = c + d; subtracting gives (a − b)(x − y) = c − d. Valid when a ≠ b and a + b ≠ 0.
8.Determinant of Order 2★
$a c b d = a d - b c$
Product along the diagonal from top-left to bottom-right minus product along the other diagonal. Put negative entries in brackets.
9.Determinant D
$D = a_{1} a_{2} b_{1} b_{2} = a_{1} b_{2} - a_{2} b_{1}$
First column = coefficients of x, second column = coefficients of y.
10.Determinant Dx
$D_{x} = c_{1} c_{2} b_{1} b_{2} = c_{1} b_{2} - c_{2} b_{1}$
Replace the x-column (first column) of D by the constants c₁, c₂.
11.Determinant Dy
$D_{y} = a_{1} a_{2} c_{1} c_{2} = a_{1} c_{2} - a_{2} c_{1}$
Replace the y-column (second column) of D by the constants c₁, c₂.
12.Cramer's Rule★
$x = \frac{D _{x}}{D}, y = \frac{D _{y}}{D}, D \neq = 0$
$D$ : Determinant of the coefficients of x and y · $D_{x}$ : D with the x-column replaced by the constants · $D_{y}$ : D with the y-column replaced by the constants
Works only when the equations are in the form a₁x + b₁y = c₁, a₂x + b₂y = c₂ and D ≠ 0.
13.When Cramer's Rule Cannot Be Used
$D = 0 \Rightarrow no unique solution$
If D = 0, the two lines are either parallel (no solution) or the same line (infinitely many solutions), so x = Dx/D cannot be found.
14.Reducible Form: Reciprocals of x and y★
$\frac{a _{1}}{x} + \frac{b _{1}}{y} = c_{1} \Rightarrow a_{1} m + b_{1} n = c_{1}, m = \frac{1}{x}, n = \frac{1}{y}$
Solve the linear equations in m and n, then x = 1/m and y = 1/n. Here x ≠ 0 and y ≠ 0.
15.Reducible Form: x + y and x − y in Denominators
$p = \frac{1}{x + y}, q = \frac{1}{x - y} \Rightarrow x + y = \frac{1}{p}, x - y = \frac{1}{q}$
After finding p and q, solve the two new linear equations x + y = 1/p and x − y = 1/q for x and y.
16.Two-Digit Number
$N = 10 x + y, N_{reversed} = 10 y + x$
$x$ : Digit in tens place · $y$ : Digit in units place
x = tens digit, y = units digit. Number + reversed number = 11(x + y); number − reversed number = 9(x − y).