Pair of Linear Equations in Two Variables - Algebraic Methods of Solution (Substitution and Elimination)

A pair of linear equations in two variables is represented by the general form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. Visually, these equations describe two straight lines on a Cartesian plane, and the solution corresponds to the point or points where these lines meet.

The Substitution Method is an algebraic technique where you solve one of the equations for one variable (e.g., $x$ in terms of $y$) and substitute that expression into the other equation. This transforms a system of two equations into a single equation with one variable.

The Elimination Method involves multiplying one or both equations by suitable non-zero constants so that the coefficients of one variable (either $x$ or $y$) become numerically equal. Adding or subtracting the equations then 'eliminates' that variable, allowing you to solve for the remaining one.

When comparing ratios of coefficients, if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines are intersecting. Visually, they cross at exactly one specific point, meaning the system is consistent and has a unique solution.

If the ratios satisfy $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel. Visually, these lines maintain a constant distance and never intersect, resulting in no solution (inconsistent system).

If all ratios are equal, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident. Visually, one line lies exactly on top of the other, meaning every point on the line is a solution, resulting in infinitely many solutions (consistent and dependent system).

Algebraic methods provide a precise way to find intersection points, especially when the coordinates are fractions or decimals that would be difficult to plot accurately on a graphical grid.