Key Points

A pair of linear equations in two variables $x$ and $y$ is represented as $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, where $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers.

Graphically, each linear equation in two variables represents a straight line. A pair of such equations is represented by two straight lines on a graph.

A pair of linear equations has exactly one unique solution if their graphs are intersecting lines. This occurs when the ratio of coefficients satisfies $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.

A pair of linear equations has no solution if their graphs are parallel lines. This occurs when the ratio of coefficients satisfies $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.

A pair of linear equations has infinitely many solutions if their graphs are coincident lines (the same line). This occurs when the ratio of coefficients satisfies $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

A system of linear equations is called consistent if it has at least one solution (unique or infinite). It is called inconsistent if it has no solution (parallel lines).

A pair of linear equations with infinitely many solutions is called a dependent pair. A dependent pair is always consistent.

For a pair of equations: (i) Intersecting lines (unique solution): $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$. (ii) Parallel lines (no solution): $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$. (iii) Coincident lines (infinite solutions): $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

An algebraic method to solve a pair of equations by expressing one variable in terms of the other from one equation and substituting this value into the second equation.

An algebraic method where one variable is eliminated by making its coefficients equal in both equations and then adding or subtracting the equations.

When solving algebraically, if you arrive at a true statement without variables (e.g., $5=5$), the system has infinite solutions. If you arrive at a false statement (e.g., $0=9$), the system has no solution.