Linear equations and inequalities

This is about equations of lines in 2-D. For lines in 3-D, see equations of lines in 3-D (HL).

Contents

• Horizontal and vertical lines
• Gradient (slope)
• Perpendicular lines
• Equations of a line
• System of two-variable linear equations
• Solving linear equations
• Solving linear inequalities

Horizontal and vertical lines

$x = 4$ is a vertical line. $y = -2$ is a horizontal line.

Gradient (slope)

The slope of a non-vertical line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is

where $\theta$ is the angle between the line and the positive $x$-axis.

When the line looks like ”/“, $m > 0$; when the line looks like ”\“, $m < 0$.

When the line is mostly horizontal $\lvert m\rvert < 1$; when the line is mostly vertical $\lvert m\rvert > 1$;

Parallel lines have the same slope.

Perpendicular lines

The exception is that vertical lines, which don’t have a defined gradient, are perpendicular to horizontal lines.

Equations of a line

Replacing a specific point $(x_2, y_2)$ with the generic point $(x, y)$ and rearranging results in the point-gradient form of a line:

Simplifying this, results in the gradient-intercept form:

which passes through the point $(0, c)$. And $c$ is the $y$-intercept.

The general form of a line is the only way to express a vertical line, as it has an undefined slope used by the other forms.

All forms are equally accepted unless specified otherwise. The point-gradient form is often the easiest to obtain. The gradient-intercept form is preached by many teachers as it’s easier to grade, but it is not required. The general form is mostly just good to know and recognize.

Only the gradient-intercept form is unique, for a given line.

System of two-variable linear equations

Your calculator can solve systems of linear equations.

Solving linear equations

All of these are, or can be solved in a similar way as, linear equations. If you do not know how to solve any of them, be sure to find out how.

YouTube: Learn how to solve a proportion by cross multiplication 

Solving linear inequalities

Linear inequalities can be solved the same way as linear equations, with the exception that

What if I must multiply or divide by an expression of $x$ and I don’t know if it’s positive or negative?

1. Don’t.
2. Try adding or subtracting instead.
3. Solve the equality then just test values on either side.