# Intermediate Algebra/Linear Equations

## Linear Equations

[edit | edit source]A **linear equation** is an equation that forms a line on a graph.

### Slope-Intercept form

[edit | edit source]A linear equation in **slope-intercept form** is one in the form such that is the slope, and is the y-intercept. An example of such an equation is:

#### Finding y-intercept, given slope and a point

[edit | edit source]The **y-intercept** of an equation is the point at which the line produced touches the y-axis, or the point where This can be very useful. If we know the slope, and a point which the line passes through, we can find the y-intercept. Consider:

Which passes through

Substitute and for and , respectively

Simplify.

Put into slope-intercept form.

#### Finding slope, given y-intercept and a point

[edit | edit source]The **slope** of a line is defined as the amount of change in x and y between two points on the line.

If we know the y-intercept of the line, and a point on the line, we can easily find the slope. Consider:

which passes through the point

Replace and with and , respectively.
Simplify.
Put into slope-intercept form.

### Standard form

[edit | edit source]The **Standard form** of a line is the form of a linear equation in the form of such that and are integers, and .

#### Converting from slope-intercept form to standard form

[edit | edit source]Slope-intercept equations can easily be changed to standard form. Consider the equation:

Subtract -3x from each side, satisfying

Multiply the entire equation by , satisfying

and are already integers, so we don't have to worry about changing them.

#### Finding the slope of an equation in standard form

[edit | edit source]In the standard form of an equation, the slope is always equal to