In this Concept, you will learn how to write equations in standard form.
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CK12 Foundation: 0502S Standard Form of Linear Equations (H264)
Try This
Now that you’ve worked with equations in all three basic forms, check out the Java applet at http://www.ronblond.com/M10/lineAP/index.html . You can use it to manipulate graphs of equations in all three forms, and see how the graphs change when you vary the terms of the equations.
Guidance
You’ve already encountered another useful form for writing linear equations:
standard form.
An equation in standard form is written
One useful thing about standard form is that it allows us to write equations for vertical lines, which we can’t do in slopeintercept form.
For example, let’s look at the line that passes through points (2, 6) and (2, 9). How would we find an equation for that line in slopeintercept form?
First we’d need to find the slope:
If we just graph the line, we can see that
Converting to Standard Form
To convert an equation from another form to standard form, all you need to do is rewrite the equation so that all the variables are on one side of the equation and the coefficient of
Example A
Rewrite the following equations in standard form:
a)
b)
c)
Solution
We need to rewrite each equation so that all the variables are on one side and the coefficient of
a)
Subtract
Add 7 to both sides to get
Flip the equation around to put it in standard form:
b)
Distribute the –3 on the righthandside to get
Add
Add 2 to both sides to get
c)
Find the common denominator for all terms in the equation – in this case that would be 6.
Multiply all terms in the equation by 6:
Subtract
Subtract 3 from both sides:
The equation in standard form is
Graphing Equations in Standard Form
When an equation is in slopeintercept form or pointslope form, you can tell right away what the slope is. How do you find the slope when an equation is in standard form?
Well, you could rewrite the equation in slopeintercept form and read off the slope. But there’s an even easier way. Let’s look at what happens when we rewrite an equation in standard form.
Starting with the equation
That means that the slope is
Example B
Find the slope and the
a)
b)
c)
Solution
a)
b)
c)
Once we’ve found the slope and
First, remember that we can also use the coverup method to graph an equation in standard form, by finding the intercepts of the line. For example, let’s graph the line given by the equation
To find the
The
To find the
The
We plot the intercepts and draw a line through them that extends in both directions:
Now we want to apply this process in reverse—to start with the graph of the line and write the equation of the line in standard form.
Example C
Find the equation of each line and write it in standard form.
a)
b)
c)
Solution
a) We see that the
We saw that in standard form
So we need to find values for
In this case, we see that multiplying
Therefore,
b) We see that the
The values of the intercept equations are already the same, so
c) We see that the
Let’s multiply the
Then we see we can multiply the
The equation in standard form is
Watch this video for help with the Examples above.
CK12 Foundation: Standard Form of Linear Equations
Vocabulary

An equation in
standard form
is written
ax+by=c , wherea,b , andc are all integers anda is positive. (Note that theb in the standard form is different than theb in the slopeintercept form.)
Guided Practice
Find the slope and the
a)
b)
Solution:
a)
b)
Explore More
For 16, rewrite the following equations in standard form.

y=3x−8 
y−7=−5(x−12) 
2y=6x+9 
y=94x+14 
y+35=23(x−2) 
3y+5=4(x−9)
For 712, find the slope and

5x−2y=15 
3x+6y=25 
x−8y=12 
3x−7y=20 
9x−9y=4 
6x+y=3
For 1314, find the equation of each line and write it in standard form.