Algebra - Simultaneous Linear Equations in Two Variables

A Simultaneous Linear Equation in two variables consists of two equations of the form $ax + by + c = 0$, where $x$ and $y$ are variables and $a, b, c$ are real numbers such that $a^2 + b^2 \neq 0$. These equations are 'simultaneous' because we seek a common pair of $(x, y)$ values that satisfy both equations at the same time.

The Method of Substitution involves expressing one variable in terms of the other from one equation and substituting this expression into the second equation. This transforms a two-variable system into a single linear equation in one variable, which is easier to solve.

The Method of Elimination involves multiplying one or both equations by suitable non-zero constants so that the coefficients of one variable become numerically equal (or opposites). By adding or subtracting the resulting equations, one variable is 'eliminated', allowing you to solve for the remaining variable.

The Method of Cross-Multiplication uses a specific ratio formula derived from the general forms $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. It follows a visual pattern where coefficients are arranged in columns to calculate $x$ and $y$ directly using products of diagonal terms.

Graphically, a linear equation in two variables represents a straight line on a Cartesian plane. When you plot two such equations, the point $(x, y)$ where the two lines intersect is the unique solution to the system. If the lines are parallel and never meet, the system has no solution.

A system of equations is 'Consistent' if it has at least one solution. Visually, this occurs when lines intersect at one point (Unique Solution) or coincide exactly (Infinite Solutions). It is 'Inconsistent' if there is no solution, which visually represents two parallel lines that never cross.

Solvability Conditions: For the equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, a unique solution exists only if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$. If the ratios of coefficients of $x$, $y$, and the constants are all equal ($\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$), the lines overlap completely, resulting in infinite solutions.