Linear Equations in Two Variables - Introduction to Linear Equations

A linear equation in two variables is an equation that can be written in the form $ax + by + c = 0$, where $a, b$, and $c$ are real numbers, and $a$ and $b$ are not both zero. This is known as the 'Standard Form'. Visually, this equation represents a straight line on a two-dimensional Cartesian plane.

The degree of a linear equation is always $1$. This means the highest power of the variables $x$ and $y$ is one. Because the degree is one, the graph of the equation does not curve or bend, appearing as a perfectly straight path that extends infinitely in both directions.

A linear equation in two variables has infinitely many solutions. Each solution is an ordered pair $(x, y)$ that makes the LHS equal to the RHS. On a coordinate plane, every single point that lies on the line representing the equation is a solution, and there are an unlimited number of such points.

To find a solution, you can assume a value for one variable (e.g., $x = 0$) and solve for the other. For instance, finding where the line crosses the vertical $y$-axis (the $y$-intercept) or the horizontal $x$-axis (the $x$-intercept) provides clear, distinct points to start drawing the line.

The solution of a linear equation is not affected when the same number is added to or subtracted from both sides of the equation, or when both sides are multiplied or divided by the same non-zero number. This allows us to rearrange equations into standard form without changing the line they describe visually.

Special cases include equations like $ax + c = 0$. While this looks like an equation in one variable, it can be written in two variables as $ax + 0y + c = 0$. Visually, if the $y$ coefficient is zero, the equation represents a vertical line. If the $x$ coefficient is zero, it represents a horizontal line.