# 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 ${\displaystyle y=mx+b}$ such that ${\displaystyle m}$ is the slope, and ${\displaystyle b}$ is the y-intercept. An example of such an equation is:
${\displaystyle y=3x-1}$

#### 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 ${\displaystyle x=0}$ 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:

${\displaystyle y=3x+b}$ Which passes through ${\displaystyle (1,2)}$
${\displaystyle 2=3(1)+b}$ Substitute ${\displaystyle 2}$ and ${\displaystyle 1}$ for ${\displaystyle x}$ and ${\displaystyle y}$, respectively
${\displaystyle 2=3+b}$ Simplify.
${\displaystyle -1=b}$
${\displaystyle y=3x-1}$ 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:

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

### Standard form

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

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

Slope-intercept equations can easily be changed to standard form. Consider the equation:
${\displaystyle y=3x-1}$
${\displaystyle -3x+y=-1}$ Subtract -3x from each side, satisfying ${\displaystyle Ax+By=C}$
${\displaystyle 3x-y=1}$ Multiply the entire equation by ${\displaystyle -1}$, satisfying ${\displaystyle A>0}$
${\displaystyle A}$ and ${\displaystyle B}$ 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 ${\displaystyle {\frac {-A}{B}}}$