# Intermediate Algebra/Linear Equations

## Linear Equations

A linear equation is an equation that forms a line on a graph.

### Slope-Intercept form

A linear equation in slope-intercept form is one in the form $y=mx+b$ such that $m$ is the slope, and $b$ is the y-intercept. An example of such an equation is:
$y=3x-1$ #### Finding y-intercept, given slope and a point

The y-intercept of an equation is the point at which the line produced touches the y-axis, or the point where $x=0$ 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:

$y=3x+b$ Which passes through $(1,2)$ $2=3(1)+b$ Substitute $2$ and $1$ for $x$ and $y$ , respectively
$2=3+b$ Simplify.
$-1=b$ $y=3x-1$ Put into slope-intercept form.

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

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:

$y=mx+4$ which passes through the point $(2,1)$ $y=mx+4$ $1=2m+4$ Replace $x$ and $y$ with $1$ and $2$ , respectively. $-3=2m$ Simplify. $-3/2=m$ $y=-3/2x+4$ Put into slope-intercept form.

### Standard form

The Standard form of a line is the form of a linear equation in the form of $Ax+By=C$ such that $A$ and $B$ are integers, and $A>0$ .

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

Slope-intercept equations can easily be changed to standard form. Consider the equation:
$y=3x-1$ $-3x+y=-1$ Subtract -3x from each side, satisfying $Ax+By=C$ $3x-y=1$ Multiply the entire equation by $-1$ , satisfying $A>0$ $A$ and $B$ are already integers, so we don't have to worry about changing them.

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

In the standard form of an equation, the slope is always equal to ${\frac {-A}{B}}$ 