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