Geometric interpretations, consistency tests, graphical solutions, algebraic methods, cross-multiplication, and conditions for solvability.
1. Linear Equations in Two Variables
An equation that can be put in the form
General Form
$$ax + by + c = 0$$
where $a, b,$ and $c$ are real numbers, and $a$ and $b$ are not both zero ($a^2 + b^2 \neq 0$), is called a linear equation in two variables $x$ and $y$.
The degree of each variable $x$ and $y$ is $1$.
Geometrically, every linear equation in two variables represents a straight line in the Cartesian plane.
2. Solution of a Linear Equation
A pair of values of $x$ and $y$, say $x = \alpha$ and $y = \beta$, is called a solution of $ax + by + c = 0$ if it makes the left-hand side equal to the right-hand side:
Condition
$$a\alpha + b\beta + c = 0$$
Key Property: A single linear equation in two variables has infinitely many solutions. Every point $(\alpha, \beta)$ on the straight line representing the equation corresponds to a valid solution.
3. Simultaneous Linear Equations in Two Variables
A pair of linear equations in the same two variables $x$ and $y$ considered together is called a system of simultaneous linear equations (or a pair of linear equations in two variables). Its standard form is:
Standard System
$$\begin{aligned}
a_1x + b_1y + c_1 &= 0 \\
a_2x + b_2y + c_2 &= 0
\end{aligned}$$
where $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers such that $a_1^2 + b_1^2 \neq 0$ and $a_2^2 + b_2^2 \neq 0$.
4. Solution of a Given System of Two Simultaneous Equations
An ordered pair $(x = \alpha, y = \beta)$ is called a solution of the system if it simultaneously satisfies both equations:
Geometrically, solving the system means finding the coordinates of the common point(s) shared by the two straight lines represented by the equations.
5. Consistent and Inconsistent Systems
Consistent System
A system of linear equations is said to be consistent if it has at least one solution.
Unique solution: Lines intersect at exactly one point.
Infinitely many solutions: Lines are coincident (dependent consistent).
Inconsistent System
A system of linear equations is said to be inconsistent if it has no solution.
Lines are parallel to each other.
There is no common point of intersection anywhere in the plane.
6. Solving Simultaneous Linear Equations: Graphical Method
To solve the system graphically, plot both straight lines $L_1$ and $L_2$ on the same Cartesian coordinate plane. Exactly one of three geometric cases must occur:
Case (i): Lines $L_1$ and $L_2$ Intersect
If the two lines intersect at a single point $P(\alpha, \beta)$, then the system is consistent and has a unique solution: $x = \alpha, y = \beta$.
Unique solution: $L_1$ and $L_2$ meet at exactly one point $P(\alpha, \beta)$.
Case (ii): Lines $L_1$ and $L_2$ are Coincident
If the two lines lie completely on top of one another, every point on the line is a common point. The system is consistent (dependent) and has infinitely many solutions.
Infinitely many solutions: every point on line $L_1 \equiv L_2$ satisfies both equations.
Case (iii): Lines $L_1$ and $L_2$ are Parallel
If the lines do not intersect at any point ($L_1 \parallel L_2$), there is no common point satisfying both equations. The system is inconsistent and has no solution.
No solution: $L_1$ and $L_2$ are parallel and never meet.
7. Algebraic Method: Substitution Method
In this method, the value of one variable is expressed in terms of the other variable from one equation and substituted into the other equation.
Steps for Substitution Method
Step 1: Choose one equation and express one variable (say $y$) in terms of the other variable ($x$):
$$y = \frac{-a_1x - c_1}{b_1}$$
Step 2: Substitute this expression for $y$ into the second equation:
$$a_2x + b_2\left(\frac{-a_1x - c_1}{b_1}\right) + c_2 = 0$$
Step 3: Solve this linear equation in one variable to get the value of $x$.
Step 4: Substitute the value of $x$ back into the expression obtained in Step 1 to determine $y$.
In this method, we eliminate one of the variables by multiplying the equations by suitable non-zero constants so that the coefficients of that variable become numerically equal.
Steps for Elimination Method
Step 1: Multiply both equations by appropriate non-zero numbers to make the coefficients of one variable (say $x$ or $y$) equal in magnitude:
$$\text{Multiply Eq. (1) by } a_2 \implies a_1a_2x + b_1a_2y + c_1a_2 = 0$$
$$\text{Multiply Eq. (2) by } a_1 \implies a_1a_2x + b_2a_1y + c_2a_1 = 0$$
Step 2: Add or subtract the two equations to eliminate the chosen variable.
Step 3: Solve the resulting single-variable equation for the remaining variable.
Step 4: Substitute this value into either original equation to find the other variable.
9. A Special Case: Interchanged Coefficients
When the coefficients of $x$ and $y$ are interchanged between the two equations:
Form
$$\begin{aligned} ax + by &= c_1 \quad \text{--- (1)} \\ bx + ay &= c_2 \quad \text{--- (2)} \end{aligned}$$
Efficient Solution Technique
Step 1 (Add the equations):
$$(a + b)x + (a + b)y = c_1 + c_2 \implies x + y = \frac{c_1 + c_2}{a + b} \quad \text{--- (3)}$$
Step 2 (Subtract the equations):
$$(a - b)x - (a - b)y = c_1 - c_2 \implies x - y = \frac{c_1 - c_2}{a - b} \quad \text{--- (4)}$$
Step 3 (Solve simple linear equations): Adding and subtracting equations (3) and (4) gives $x$ and $y$ directly without working with large coefficients.
10. Method of Cross-Multiplication
For the standard simultaneous system written with all terms on the left:
11. Conditions for Solvability of Linear Equations
For the system $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, comparing coefficient ratios establishes the nature of the solution without solving the equations: