EQUATIONS OF LINES
1. Writing Equations of Lines
There are many ways to define a line, but for today, let’s think of a LINE as a collection
of points such that the slope between any two of those points is the same.
You can write the equation of a line using either of the following two formulas:
SLOPE/INTERCEPT
y = mx + b
POINT/SLOPE
y – y1 = m(x – x1)
Note: It doesn’t matter which formula you use, but it’s generally easier to use the
formula named after the information you know about your line:
•
If you know the slope and a point, then use POINT/SLOPE
•
If you know the slope and the y-intercept, then use SLOPE/INTERCEPT
When the question says, “Write the equation of the line . . .”
STEP 1: Ask
Do I know the SLOPE of the line? (Notice how it’s needed for both!)
YES!
NO! But, I have two
points on the line.
NO! But, my line is
parallel/perpendicular
to another given line.
Good! Write
down m =
Not bad! Calculate
the slope using:
m = y2 – y1
x2 – x 1
then write down m =
That’s OK! Find the
slope of THAT line
by rearranging into
y = mx + b so you can
see its slope
PERPENDICULAR PARALLEL
Use the “negative
Use the exact
reciprocal” slope
same slope
Then write down m =
STEP 2: Ask
Do I know the Y-INTERCEPT of the line?
YES!
Good, write b =
NO! But I know a different point instead.
OK, write x1= and y1=
USE y = mx + b
Fill in m, and b, and you’re done!
y – y1 = m(x – x1)
Fill in m, x1 and y1 then rearrange into y = mx+b
Let’s try some examples . . .
Write the equation of the line that
(a) passes through the point P(2, 3) and has a slope of 4
(b) has a slope of -3 and a y-intercept of 12
(c) passes through (1, 4) and (2, 8)
(d) passes through (-2, -5) and is parallel to 4x – 3y = 8
(e) passes through (1, 7) and is perpendicular to x – 2y – 3 = 0
Let’s try some examples . . .
Write the equation of the line that
(a) passes through the point P(2, 3) and has a slope of 4
(b) has a slope of -3 and a y-intercept of 12
(c) passes through (1, 4) and (2, 8)
(d) passes through (-2, -5) and is parallel to 4x – 3y = 8
(e) passes through (1, 7) and is perpendicular to x – 2y – 3 = 0
Note to teacher/parent: Students often have difficulty with the notion of lines written in “standard form” because, unlike
the other formula they know (POINT/SLOPE and SLOPE/INTERCEPT) they cannot begin a question using this formula.
Yet, many textbook questions are simply phrased, “Write the equations of the following lines in standard form,” allowing
students to forget that they must use a different form to get the question started! It must be emphasized that one arrives at
standard form through a process of steps and that standard form is really just a conventional notation for their final
answer.
What about STANDARD FORM? What’s up with that formula? When do I use it?
STANDARD FORM is simply a standard way of expressing your final answer.
You do not begin a question by using standard form!
You get there by first using one of the earlier two formulas and then rearranging!
It looks like this: Ax + By + C = 0 or like this:
0 = Ax + By + C
But, unlike the other two formulas we use, the A, B, and C themselves don’t represent
anything on the graph. (You’ll notice, over time, that combinations of A, B, and C have
meaning, but based on what we know now, A, B and C don’t mean very much.)
Remember, m represents the slope of the line, b represents the y-intercept of the line
while x1 and y1 represent the x and y values of a specific point on your line. Some
combination of these properties of the graph will be given to you in the question, making
it very easy to “plug into” either SLOPE/INTERCEPT or POINT/SLOPE to get an
equation. So, you should begin your question with one of these two formula in mind.
You see, you don’t get values for A, B and C from your question
you simply rearrange
one of the other two formulas until it looks like Ax + By + C = 0. This “formula” is
really just meant to be a guide showing you what your final answer should look like.
Specifically:
You’ll know your equation is in standard form when it satisfies the following criteria
1. All terms must be moved to one side so that one side is left “empty.” ( = 0 )
2. The terms must be written in alphabetical order according to the variables. (the x
term comes first, then the y term, then the constant)
3. There can be no fractions or decimals. (multiply the entire equation or use
common denominators to clear fractions)
4. The x term must be positive. (If necessary, multiply/divide by -1 throughout
OR move all terms to the other side of the equation so that x is positive)
Let’s convert our answers from the previous examples into standard form:
y = 4x – 5
0 = 4x – 5 – y
0 = 4x – y – 5
Why is this not standard form?
Why is this not standard form?
needs to = 0
needs terms in proper order
y = -3x +12
0 = -3x +12 – y
0 = -3x – y +12
0 = 3x + y – 12
Why is this not standard form?
Why is this not standard form?
Why is this not standard form?
needs to = 0
needs terms in proper order
needs positive coefficient of x
How about some more?
needs to = 0
needs terms in proper order
no fractions allowed
y = ½x – ¾
0 = ½x – ¾ - y
0 = ½x – y – ¾
Why is this not standard form?
Why is this not standard form?
Why is this not standard form?
0 = ½x – y – ¾
*use your preferred method for “clearing fractions” usually one of:
Multiply each term by the common
Rewrite every term with a common
denominator (on both sides, including OR denominator (on both sides, including the 0)
the 0). Then, reduce/cancel all
and then simply cancel out the denominators
individual fractions separately
0 = ½x – y – ¾
0 (4) = ½x(4) – y(4) – ¾(4)
0 = 2x – 4y – 3
0 = ½x – y – ¾
0/4 = 2/4 x – 4/4 y -3/4
0 = 2x – 4y – 3
needs to = 0
y = -3/4x + 2/3
Why is this not standard form?
0 = -3/4x + 2/3 – y Why is this not standard form?
needs terms in proper order
0 = -3/4x – y + 2/3 Why is this not standard form?
no fractions allowed
0/12 = -9/12 x – 12/12 y + 8/12
OR
0(12) = -3/4x(12) – y(12) + 2/3(12)
0 = -9x – 12y + 8
Why is this not standard form?
needs positive coefficient of x
Move terms to other side
9x + 12y – 8 = 0
9x + 12y – 8 = 0
OR
OR
OR
Multiply or Divide every term by -1
0(-1) = -9x(-1) – 12y(-1) + 8(-1)
0 = 9x + 12y – 8
needs to = 0
needs terms in proper order
needs positive coefficient of x
y = 9 – 3x
0 = 9 – 3x – y
0 = -3x – y + 9
0 = 3x + y – 9
Why is this not standard form?
Why is this not standard form?
Why is this not standard form?
2x – 4y = 7
2x – 4y – 7 = 0
Why is this not standard form?
needs to = 0
(Notice, it doesn’t matter which side = 0)
In Conclusion:
•
Standard form is a way of expressing your final answer
•
You must first have an equation of a line before you can put it in standard form
•
You’ll know your answer is in standard form when it satisfies the checklist:
o One side equals zero
o Terms are in proper order (x, then y, then constant)
o No fractions or decimals
o Starts with a positive coefficient of x
•
Some equations will take more work than others to put in standard form
•
When you begin to notice patterns, then you can take shortcuts and combine steps
SUMMARY
Now you should be able to write the equation of a line when given …
o A point and the slope
(You have x1 and y1 and m, so use y – y1 = m(x – x1)
o The slope and the y-intercept
(You have m and b so use y = mx + b
o Two points
(You have x1, y1, x2 and y2
so use m = y2 – y1 / (x2 – x1) and then y = mx + b
o A point and told you are parallel to another given line
(You have x1 and y1 and you will have m when you rearrange the given line into
y = mx + b and use the same slope as that line so use y – y1 = m(x – x1)
o A point and told you are perpendicular to another given line
(You have x1 and y1 and you will have m when you rearrange the given line into
y = mx + b and use the negative reciprocal slope so use y – y1 = m(x – x1)
And, you can then rearrange any of these equations into standard form Ax + By + C = 0
which requires that . . .
o One side equals zero
o Terms are in proper order (x, then y, then constant)
o No fractions or decimals
o Starts with a positive coefficient of x
2. Finding x- and y-intercepts
Remember: an intercept is the value where the line crosses through the axis.
The x-intercept is the value of x where the line crosses the x-axis.
The y-intercept is the value of y where the line crosses the y-axis.
Note: The intercepts occur
where the other
variable equals zero
So, to find intercepts, we take turns setting each variable equal to zero.
Example: 4x + 3y = 12
For the x-intercept, let y = 0
TRICK! Cover up the entire
y term with your finger because
it’s zero it’s gone!
4x = 12
x = 3
For the y-intercept, let x = 0
TRICK! Cover up the entire
x term with your finger, because
It’s zero it’s gone!
3y = 12
y = 4
Therefore, the x-intercept is 3 and the y-intercept is 4.
This means that the line crosses the x-axis at 3 and the y-axis at 4.
Example:
2x – 5y = 20
For the x-intercept
let y = 0
(cover up y!)
2x = 20
x = 10
For the y-intercept
let x = 0
(cover up x!)
-5y = 20
y = -4
Example:
½ x – 3y = 12
For the x-intercept
let y = 0
(cover up y!)
½ x = 12
x = 24
For the y-intercept
let x = 0
(cover up x!)
-3y = 12
y = -4
Example:
-6x + 5y = 90
For the x-intercept
let y = 0
(cover up y!)
For the y-intercept
let x = 0
(cover up x!)
-6x = 90
x = -15
5y = 90
y = 18
Example:
4x – 5y – 20= 0
For the x-intercept
let y = 0
(cover up y!)
4x – 20 = 0
4x = 20
x=5
For the y-intercept
let x = 0
(cover up x!)
– 5y – 20= 0
– 5y = 20
y = -4
3. Graphing Lines
This brings us to graphing lines when given their equations.
There are 3 common ways to graph a line
(You can usually choose the method you prefer for each individual question)
1. Intercept Graphing
a. Find both intercepts
b. Plot both intercepts
c. Connect the two points with a straight line
Let’s use intercept graphing for the equations on the previous page:
Example:
2x – 5y = 20
For the x-intercept
let y = 0
(cover up y!)
2x = 20
x = 10
For the y-intercept
let x = 0
(cover up x!)
-5y = 20
y = -4
Example:
½ x – 3y = 12
For the x-intercept
let y = 0
(cover up y!)
½ x = 12
x = 24
For the y-intercept
let x = 0
(cover up x!)
-3y = 12
y = -4
Example:
-6x + 5y = 90
For the x-intercept
let y = 0
(cover up y!)
For the y-intercept
let x = 0
(cover up x!)
-6x = 90
x = -15
5y = 90
y = 18
Example:
4x – 5y – 20= 0
For the x-intercept
let y = 0
(cover up y!)
4x – 20 = 0
4x = 20
x=5
For the y-intercept
let x = 0
(cover up x!)
– 5y – 20= 0
– 5y = 20
y = -4
2. SLOPE/INTERCEPT Graphing using y = mx + b
a. Make sure your equation is written in y = mx + b form
b. Plot the b value (remember, this is just the y-intercept)
c. Write your slope as a fraction so you can clearly see the rise and run
Examples: y = 2x – 4
y = 0.5x – 7
y = -x/5 + 8
d. From the y-intercept you plotted, move according to your slope (do the
“rising” and the “running”) and then plot a second point where you end up
e. Connect the two points with a straight line
Let’s plot the previous examples using y = mx + b
Example:
2x – 5y = 20
Example:
-6x + 5y = 90
Example:
½ x – 3y = 12
Example:
4x – 5y – 20= 0
3. Graphing using a Table of Values
a. Make sure your equation is written in y = mx + b form
b. Draw a “Table of Values” as shown below
c. Choose values for x (you can really choose any you want, but if you’re not
sure, choose -3, -2, -1, 0, 1, 2, 3 until you are comfortable picking values)
d. Sub each of the the x-values into the equation to get their corresponding
y-values. Note, each pair of x and y values is really a point on your graph.
e. Plot the points and connect them to make a straight line.
Let’s plot the previous examples using tables of values
Example:
2x – 5y = 20
Example:
-6x + 5y = 90
Example:
½ x – 3y = 12
Example:
4x – 5y – 20= 0
Graph the following using a method of your choice.
(a) y = 3/2x – 4
(b) x + 2y = 6
(c) 5x – 2y = 10
Graph the following using a method of your choice.
(a) y = 3/2x – 4
(b) x + 2y = 6
(c) 5x – 2y = 10
m=0
m = undefined
Don’t forget about HORIZONTAL and VERTICAL lines!
(We don’t even need a formula because they’re so easy!)
You’ll recognize them because they only contain one variable, for example:
y=2
x=5
When you see these equations, remind yourself that they must be pretty simple, because
there’s only one variable. Instead of freaking out because they look “different,” just take
a moment to remember that we have a different strategy for equations that are so easy.
These equations are simply telling you what will always be true for any point on the line.
STRATEGY: Just find a few points that satisfy the condition, then plot them!
y=2
List some points where y = 2
(0, 2) (3, 2) (-4, 2) (6, 2)
x=5
List some points where x = 5
(5, 0) (5, 3) (5, -2) (5, 1)
Can you do it backwards? Can you write the equations of the lines graphed below simply
by checking to see what condition is always true?