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The slope formula is an equation with a fraction (๐ =
โ๐ฆ
โ๐ฅ
). To eliminate
the fraction, we can multiply both sides of the equation by the
denominator.
This means the vertical change
of a line is equal to its slope times its
horizontal change
. This leads us to what is known as point-slope
form , which is a way to find the equation of a line using any point that the
line passes through and the slope of the line.
Point-slope form:
- โ๐ฆ = ๐ โ โ๐ฅ or ๐ฆ โ ๐ฆ
1
1
- a formula used to find the equation of a line that is passing through a
point ๐
1
1
and that has a slope ๐
o plug in the ๐ฅ-coordinate of the point for ๐ฅ
1
, plug in the ๐ฆ-
coordinate of the point for ๐ฆ
1
, plug in the slope of the line for ๐,
and then simplify completely removing parentheses and
combining like terms
- when using this formula we will always leave ๐ฅ and ๐ฆ as ๐ฅ and ๐ฆ and
only enter values for ๐ฅ
1
1
, and ๐
Example 1 : Find the equation of a line that passes through the point
and has a slope of โ
3
4
. Isolate the ๐ฆ variable in the equation.
We have a point that the line passes through
and the slope of the
line (๐ = โ
3
4
), so we can use point-slope form to find the equation of
the line.
1
1
Example 3 : Find the equation of the line that passes through the point
( 3 , โ 2 ) and has a slope 5. Enter exact answers only (no approximations),
and write the equation in slope-intercept form
We have a point that the line passes through
and the slope of the
line
, so we can use point-slope form to find the equation of the
line.
1
1
Example 4 : Find the equation of the line that has a ๐ฆ-intercept of โ
2
5
and
has a slope
1
7
. Enter exact answers only (no approximations), and write
the equation in slope-intercept form
Example 5 : Find the equation of the line that has an ๐ฅ-intercept of โ
2
๐
and a slope of ๐. Enter exact answers only (no approximations), and
write the equation in slope-intercept form (๐ฆ = ๐๐ฅ + ๐).
To find the equation of a line, you need to have one point that the line
passes through and the slope of the line; this is how we found the
equations of the lines on Examples 1 โ 5. If you do not have the slope of
the line, you will need two points that the line passes through so you can
find the slope; this is what we will do on Examples 6 and 7.
Once you have the slope of the line, and at least one point that the line
passes through, you can use point-slope form (๐ฆ โ ๐ฆ
1
1
)) to
find the equation of the line. Point-slope form was used (or will be used)
on each example except Example 4, when slope-intercept form was used
since we had the slope of the line and the ๐ฆ-intercept.
Example 7 : Find the equation of the line that has an ๐ฅ-intercept of โ 3 and
a ๐ฆ-intercept of 4. Enter exact answers only (no approximations), and
write the equation in slope-intercept form
Answers to Examples:
3
4
31
4
5
2
21
2
1
7
2
5
6
5
17
5
4
3
