Linear Equations
Slope
Definition of Slope 
The steepness of a line in the coordinate
plane is called its slope. It is defined
as the ratio of the rise, or vertical change in y, to the
run, or horizontal change as you move from one point to
the other. 
Determining Slope Given Two
Points 
Given the coordinates of two points, ( x_{1},
y_{1}) and ( x_{2}, y_{2}), on a
line, the slope m of the line can be found as follows. 
Example
What is the slope of the line that passes through (4, 6) and
(2, 3)?
Solution
Let x_{1} = 4, y_{1 }= 6, x_{2 }= 2,
and y_{2 }= 3.
Writing Linear Equations in PointSlope and Standard Forms
PointSlope Form of a Linear
Equation 
For a given point ( x_{1}, y_{1})
on a nonvertical line having slope of m, the pointslope
form a linear equation is as follows: y  y_{1}=
m( x  x_{1}). The linear equation of a
vertical line, which has an undefined slope, through a
point ( x_{1}, y_{1}) is x = x_{1}.

Standard Form 
The standard form of a linear equation is
Ax + By = C, where A, B, and C are integers, A 0, and A and B are not both zero. 
Example
Write the equation, first in pointslope form and then in
standard form, of the line that passes through (2, 3) and has a
slope of 5.
Solution
Point Slope Form 
y  y_{1 }= m( x  x_{1}) 

y  3 = 5( x  2) 
y  3 = 5x  10 
Distribute. 
5x  10 = y  3 
Reflexive Property ( = ) 
5x  y = 7 
Add 10 and subtract y from each side. 
Standard Form: 
5x  y = 7, where A = 5, B = 1 and C = 7 
Writing Linear Equations in SlopeIntercept Form
The coordinates at which a graph intersects the axes are known
as the xintercept and the yintercept.
Finding Intercepts 
To find the xintercept, substitute 0 for
y in the equation and solve for x. To find the
yintercept, substitute 0 for x in the equation and solve
for y. 
SlopeIntercept Form of a Linear
Equation 
If a line has a slope of m and a
yintercept of b, then the slopeintercept form of an
equation of the line is y = mx + b. 
Example
Find the x  and y intercepts of the graph of 2x + 3y = 5.
Then, write the equation in slopeintercept form.
Solution
2x + 3(0) = 5 
Let y = 0 
2(0) + 3y = 5 
Let x = 0 
2x = 5 

3y = 5 


The xintercept is 

The yintercept is 
SlopeIntercept form: 2x + 3y
= 5 


