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, ( x1,
y1) and ( x2, y2), 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 x1 = 4, y1 = -6, x2 = -2,
and y2 = 3.
Writing Linear Equations in Point-Slope and Standard Forms
Point-Slope Form of a Linear
Equation
For a given point ( x1, y1)
on a nonvertical line having slope of m, the point-slope
form a linear equation is as follows: y - y1=
m( x - x1).
The linear equation of a
vertical line, which has an undefined slope, through a
point ( x1, y1) is x = x1.
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 point-slope 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 - y1 = m( x - x1)
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 Slope-Intercept Form
The coordinates at which a graph intersects the axes are known
as the x-intercept and the y-intercept.
Finding Intercepts
To find the x-intercept, substitute 0 for
y in the equation and solve for x. To find the
y-intercept, substitute 0 for x in the equation and solve
for y.
Slope-Intercept Form of a Linear
Equation
If a line has a slope of m and a
y-intercept of b, then the slope-intercept 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 slope-intercept form.
Solution
2x + 3(0) = 5
Let y = 0
2(0) + 3y = 5
Let x = 0
2x = 5
3y = 5
The x-intercept is
The y-intercept is
Slope-Intercept form: 2x + 3y
= 5
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