Writing Equations of Lines
Starting with Point – Slope Form
y – y1 = m (x – x1)
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DONE
Point – Slope Form
To write an equation of a line in point – slope form, all you need is …
… Any Point On The Line … (x1, y1)
… The Slope …
m
Once you have these two things, you can write the equation as
y – y1 = m (x – x1)
That’s “y minus the y-value of the point equals the slope times the
quantity of x minus the x-value of the point”.
Note: This equation is not in function form … more on that later.
DONE
Example #1
Write the equation of the line that goes through the point (2, –3) and has a slope of 4.
Point = (2, –3)
Slope = 4
Starting with the point – slope form
y – y1 = m (x – x1)
Plug in the y-value, the slope, and the x-value to get
y + 3 = 4 (x – 2)
Notice, that when you subtracted the “–3” it became “+3”.
DONE
Example #2
Write the equation of the line that goes through the point (–4, 6) and has a slope of
3
.
2
Point = (–4, 6)
3
Slope =
2
Starting with the point – slope form
y – y1 = m (x – x1)
Plug in the y-value, the slope, and the x-value to get
y–6=
3
(x + 4)
2
Notice, that when you subtracted the “–4” it became “+4”.
DONE
Example #3
Write the equation of the line that goes through the points (6, –4) and (2, 8) .
We have two points, but we’re missing the slope. Using the formula for slope, we can
find the slope to be
y2 – y1
8- (- 4) 12
m=
=
=- 3
x2 – x1
2- 6
- 4
To use point – slope form, we need a point and a slope. Since we have two points,
just pick one … IT DOESN’T MATTER … BOTH answers are acceptable… more
on why later.
Using the first point, we have,
Using the second point, we have,
Point = (6, –4)
Point = (2, 8)
Slope = –3
Slope = –3
y + 4 = –3 (x – 6)
y – 8 = –3 (x – 2)
DONE
Other Forms of Linear Equations
So far, we have discussed only point-slope form. There are other forms of equations
that you should be able to identify as a line and graph if necessary.
Horizontal Line:
y = c , where c is a constant.
Vertical Line: x = c , where c is a constant.
Slope – Intercept Form:
Example:
y=3
Example:
x = –6
y = mx + b
m = the slope of the line … b = the y-intercept
Example:
y = 3x – 6
Standard Form: Ax + By = C
A, B, and C are integers.
Example:
3x + 4y = –36
To write equations in the
last two forms, start in
point – slope form and
rearrange the variables to
match the correct format.
The next few slides will
cover how to do this.
DONE
Writing Equations in Slope – Intercept Form
Earlier (click here to review) we wrote an equation of the line that went through the
points (6, –4) and (2, 8) . Sometimes, we want the line written in a different form.
To change a point-slope equation in slope-intercept form, solve for y and simplify the
right side of the equation.
- Solve for y: Add or subtract the y-value of the point to both sides
- Simplify: Distribute the slope and then combine like terms.
Here are the two answers we had from the earlier example.
y + 4 = –3 (x – 6)
y – 8 = –3 (x – 2)
SOLVE FOR y
Subtract 4 from both sides
Add 8 to both sides
y = –3 (x – 6) – 4
y = –3 (x – 2) + 8
Distribute –3 and combine like terms
SIMPLIFY
Distribute –3 and combine like terms
y = –3x + 18 – 4
y = –3x + 6 + 8
y = –3x + 14
y = –3x + 14
Notice … They’re the same!
DONE
Example #4
Write the equation of the line in slope-intercept form that goes through the point (6, 2)
2
and has slope
.
3
Begin in point-slope form:
y–2=
2
(x – 6)
3
Add 2 to both sides
Solve for y:
y=
2
(x – 6) + 2
3
2
x–4+2
3
Distribute:
y=
Combine Like Terms:
2
y= x–2
3
DONE
Writing Equations In Standard Form
The last form of a linear equation we are going to cover is called Standard Form.
Ax + By = C , where A, B, and C are integers.
If you needed to write an equation of a line in standard form, you would start in
point-slope form or slope-intercept form, depending on what information you are
given.
In both cases, you must put all variables on the left side and all constant values on
the right side.
If any of the coefficients (A, B, or C) are NOT integers, then you must eliminate
any fractions or decimals by multiplying every term in the equation by the
appropriate factor.
DONE
Example #5
Rewrite each of the equations below in standard form.
y=
Subtract
2
x–4
3
2x
from both sides.
3
- 2 x+ y = - 4
3
Multiply ALL terms by 3 in
order to eliminate the fraction.
–2x + 3y = –12
3
(x + 4)
2
3
Distribute
2
y- 6 = 3 x + 6
2
Subtract 3 x from both sides,
2
and add 6 to both sides.
y–6=
- 3 x + y = 12
2
Multiply ALL terms by 2.
– 3x + 2y = 24
DONE
y as a FUNCTION of x
For an equation to be written as a function, you must solve for y.
Solving for y means that “y is written as a function of x ”.
Of the three types of linear equations discussed in this presentation, only slopeintercept form is written as a function.
When your equation is in point – slope form simply add or subtract the y-value of
the point to the other side.
From our first example we had
y + 3 = 4 (x – 2)
In order to write y as a function of x we subtract 3 from both sides of the equation.
y = 4 (x – 2) – 3
When you write y as a function of x, you have put your equation in function form.
You may replace the y with the notation f (x) … read “f of x ” or “function of x ”.
f (x) = 4 (x – 2) – 3
DONE