# 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 ${\frac {-A}{B}}$