Finding the equation of a line algebraically - Purdue University

16-week Lesson 16 (8-week Lesson 12)

Finding the equation of a line algebraically

??

The slope formula is an equation with a fraction (? = ?? ). 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:

- ?? = ? ? ?? or ? ? ?1 = ?(? ? ?1 )

- a formula used to find the equation of a line that is passing through a

point ?(?1 , ?1 ) and that has a slope ?

o plug in the ?-coordinate of the point for ?1 , plug in the ?coordinate of the point for ?1 , plug in the slope of the line for ?,

and then simplify completely removing parentheses and

combining like terms

- when using this formula we will always leave ? and ? as ? and ? and

only enter values for ?1 , ?1 , and ?

1

16-week Lesson 16 (8-week Lesson 12)

Finding the equation of a line algebraically

Example 1: Find the equation of a line that passes through the point (1,7),

3

and has a slope of ? 4. Isolate the ? variable in the equation.

We have a point that the line passes through (1,7) and the slope of the

3

line (? = ? 4), so we can use point-slope form to find the equation of

the line.

? ? ?1 = ?(? ? ?1 )

3

? ? 7 = ? (? ? 1)

4

3

3

??7=? ?+

4

4

3

3

? =? ?+ +7

4

4

?

??

?=? ?+

?

?

2

16-week Lesson 16 (8-week Lesson 12)

Finding the equation of a line algebraically

The equation we found in Example 1 was written as ? in terms of

3

31

(? = ? 4 ? + 4 ). That means that the ? variable is dependent on the ?

variable. This is what is known as slope-intercept form, where the

coefficient of the ? variable is the slope and the constant term is the ?intercept. Looking at the equation from Example 1, the slope of that line

3

31

is ? 4 and the ?-intercept is 4 .

Slope-intercept form:

- ? = ?? + ?

- a common way of writing the equation of a line that identifies the

slope ? and the ?-intercept (0, ?)

- the result of solving a linear equation for the variable ?

o this means expressing ? in terms of ?

Example 2: Find the equation of a line that passes through the point

5

(?5, ?2), and has a slope of . Enter exact answers only (no

2

approximations), and write the equation in slope-intercept form

(? = ?? + ?).

We have a point that the line passes through (?5, ?2) and the slope of

5

the line (? = 2), so we can use point-slope form to find the equation of

the line.

? ? ?1 = ?(? ? ?1 )

5

? ? (?2) = (? ? (?5))

2

5

? + 2 = (? + 5)

2

5

25

?+2= ?+

2

2

?

??

?= ?+

?

?

3

16-week Lesson 16 (8-week Lesson 12)

Finding the equation of a line algebraically

Example 3: Find the equation of the line that passes through the point

(3, ?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 (3, ?2) and the slope of the

line (? = 5), so we can use point-slope form to find the equation of the

line.

? ? ?1 = ?(? ? ?1 )

? ? (?2) = 5(? ? 3)

? + 2 = 5? ? 15

? = ?? ? ??

2

Example 4: Find the equation of the line that has a ?-intercept of ? 5 and

1

has a slope 7. Enter exact answers only (no approximations), and write

the equation in slope-intercept form (? = ?? + ?).

??????? ????

????? ? ????? ????:

(?? , ?? )

? ? ?? = ? ( ? ? ?? )

(

,

)

? ? ??

4

=

?

( ? ? ??

)

16-week Lesson 16 (8-week Lesson 12)

Finding the equation of a line algebraically

2

Example 5: Find the equation of the line that has an ?-intercept of ? ?

and a slope of ?. Enter exact answers only (no approximations), and

write the equation in slope-intercept form (? = ?? + ?).

??????? ????

????? ? ????? ????:

(?? , ?? )

? ? ?? = ? ( ? ? ?? )

?

(? ?

,? )

? ? ??

=

?

(? ?

?

? ?)

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 �C 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 (? ? ?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 ?-intercept.

5

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