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If we know the intercepts and the slope of a linear equation, we know everything there is to know about it. Sure, we may not know where it was born, who its favorite musical artist is, or its stance on marriage equality, but for our purposes, we know it well enough.
There are different ways to package this information. That is, the same linear equation can be written in multiple ways.
This fact shouldn't come as too much of a surprise, so wipe that "surprised face" off your mug. After all, we already know that equations can be rearranged to produce equivalent equations. The following equations are all equivalent to each other (check and see):
These equations are in the three most common forms used for writing linear equations:
For each of these forms, we'll talk about how to go from the equation to the graph, and how to go from the graph to the equation. It isn't simply a matter of hitting "Reverse Directions" on your GPS.
A linear equation in standard form is an equation that looks like
ax + by = c
where a, b, and c are real numbers and a and b aren't both zero. But c can be zero if it wants. It's the favorite child, so it gets special privileges.
If only a = 0, the equation can be rewritten to look like this:
y = (some number)
If only b = 0, the equation can be rewritten, too:
x = (some number)
For example, the equation 8y = 3 is equivalent to the equation , which is also in standard form (with b = 1).
Meanwhile, the equation 2x = 4 is equivalent to the equation x = 2, which is also in standard form (with a = 1).
If either a or b is zero, we know how to graph the equation and how to read off an equation from a graph. You probably suspect there will be some cases where it won't be so easy, and neither a nor b will be zero. You suspect right.
Okay, now what if an equation throws us a curveball? Should we sacrifice our bodies and take our base?
If neither a nor b is zero, we can most easily graph the linear equation by finding its intercepts.
Graph the linear equation x + 4y = 8.
Let's find the intercepts. To find the x-intercept, let y = 0 and solve for x, since the x-intercept will be at a point of the form (something, 0).
x + 4(0) = 8
So x = 8 is the x-intercept.
For the y-intercept, let x = 0 and solve for y.
0 + 4y = 8
And y = 2 is the y-intercept. Sweet, we've tracked down both intercepts. Who needs a or b to be zero? Not us.
Now we can plot the intercepts:
Connect the dots to get the line:
Write, in standard form, the linear equation graphed below:
The x intercept is at (-1, 0), which means whatever a, b, and c are, our equation looks like this:
a(-1) + b(0) = c
Let's make life easy on ourselves and let a = 1. That's right...we're going to dip this equation in a bucket of A-1 sauce.
1(-1) + b(0) = c
-1 = c
To find b, the remaining coefficient, we look at the y-intercept: y = -2. At that point, x will be 0, and we've already decided that c = -1, so we find:
0 + b(-2) = -1
Therefore, . We now know all the coefficients and can write the equation:
If we want to make things pretty, we can multiply both sides of the equation by 2 and write the resulting equation, which has integer coefficients. If we want to make things really pretty, we can dress the equation up in a sequined ball gown and give it a makeover. Let's start small, though:
2x + y = -2
Write, in standard form, the linear equation graphed below:
The x intercept is -2, which means whatever a, b, and c are, our standard-form equation is:
a(-2) + b(0) = c
We can let a = 1, so:
-2 = c
To find b we look at the y-intercept, which occurs at (0, 4). And since we've decided c = -2, we find:
0 + b(4) = -2
This means . We now know all the coefficients. Not on a first-name basis, but well enough to get by. We can now write the equation.
To make things pretty, we can multiply both sides of the equation by 2 to get an equivalent equation with integer coefficients:
2x – y = -4
Now for that makeover.
A linear equation in slope-intercept form looks like
y = mx + b
where m and b are real numbers and m is the slope of the line. Get to know slope-intercept form, because it's going to be one of your new best friends. You two are going to have sleepovers, do each other's hair, and play Fantasy Date. Then slope-intercept form is going to go behind your back and kiss Davy Phillips, and it's going to cause a whole big thing. We're getting ahead of ourselves.
When x = 0, here's what happens to slope-intercept form:
y = m(0) + b
y = b
Guess what that means: b is the y-intercept (the value of y when x = 0).
To graph a line in slope-intercept form, we graph the y-intercept and then use the slope to plot another point or two. Or seven, if you have the time.
Graph the line y = 2x + 5.
When x = 0, y = 5, so the y-intercept is at 5.
The slope is 2, so as x moves right by 1, y moves up by 2 to give us the point (1, 7):
Then we connect the dots:
To check our work, we can plug in some other value of x and make sure the point we get is actually on the line. If the point was on the line but is no longer, it probably has AT&T.
Plug in x = -1:
y = 2(-1) + 5 = 3
So we should have the point (-1, 3) on the graph. Thankfully, we do.
To read off the equation of a line in slope-intercept form, we need to figure out the y-intercept, the slope of the line, and the z-intercept. Psych! There is no z-intercept. We're just checking to see if you're paying attention.
Find the equation of the line graphed below.
We can see from the picture that the y-intercept is 2, so the equation for the line is:
y = mx + 2
Now we need to figure out m, the slope of the line. We can see from the picture that as x moves right by 1, y increases by 3.
The slope of the line is m = 3, so the final equation for the line is:
y = 3x + 2
We can also find the equation of a line in slope-intercept form if we're given two points but we don't know the y-intercept. First we find the slope, then we find the y-intercept by sneaky methods. Put on your black mask and grab your throwing stars. We're going ninja.
Find the equation of the line in slope-intercept form.
First, we find the slope. Slope is , which we can figure out from the graph:
The slope is . Slope is m in the equation y = mx + b, so we know the equation for this line is:
All we need to do is figure out b. Someone borrowed our copy of the Big Book of What Variables Equal, so we'll have to do this manually. Since we know the points (1, 4) and (10, 7) are on this line, each of them must satisfy the equation. We can use one point to find b, and the other to check that we're right.
To find b, we know the point with x = 1 and y = 4 must be on the line, so let's plug those values into the equation:
Now we solve for b.
The equation for the line should be .
To make sure we're right, let's check that the other point (the one with coordinates x = 10 and y = 7) satisfies this equation. If not, maybe we can buy it a steak.
The left-hand side of the equation is 7, and the right-hand side is , which is also 7. Both the points (1, 4) and (10, 7) satisfy this equation, so we found the right equation for the line. Oh, good. We didn't want to have to shell out for that steak.
A line is in point-slope form if it looks like
y – y1 = m(x – x1)
where y1, x1, and m are real numbers. Here (x1, y1) is a fixed point on the line, and m is the slope of the line. In fact, (x1, y1) is so fixed that it's never going to birth a litter. #petjokes
To graph an equation given in point-slope form, it's often easiest to rewrite the equation in slope-intercept form.
Graph the equation y – 3 = 4(x – 0.5).
First we add 3 to each side:
y = 4(x – 0.5) + 3
Then simplify to get:
y = 4x + 1
From here, we can graph the equation using the y-intercept and the slope:
Point-slope form is most useful for finding the equation of a line when you're given either a graph or two points on the line. By the way, when you're given a graph, say "thank you" and don't ask for any more. You don't want to look a gift graph in the mouth.
Find the equation of the line shown below.
First we need to pick a point (x1, y1). Let's take a point with nice, even integer coordinates. Yes, 14,838 and 372,410 are even numbers, but we can do better. Let (x1, y1) be the point (0, 1), so x1 = 0 and y1 = 1.
Now we need to find the slope, m, of the line. Pick another point on the line and look at the rise and run between the two points. Don't look at anything else if you can help it; this slope is a little self-conscious.
We can conclude that .
To write the equation for the line, we use the blueprint y – y1 = m(x – x1) and plug in the values x1 = 0, y1 = 1, and .
Rearrange that bad boy to get:
Here's a fun trick (and yeah, we're using "fun" very, very loosely): if we rearrange the point-slope equation y – y1 = m(x – x1), we find:
If we fix a point (x1, y1) on the line, then for any other point (x, y) on the line we can think of y – y1 as the rise and x – x1 as the run. We know how much you love your visual aids, and we would never dream of depriving you of them, so here you go:
Since m is the slope of the line, saying is really just saying , which we know is true. And just like that, we've got a handy new formula for finding the slope.
If you're not told specifically which form of a linear equation to use, use whichever you like. Yep, seriously. If you're asked to graph an equation in standard form but you like point-slope better, you can turn the equation into point-slope form and then graph it. If you're asked to figure out an equation from a graph and not told how, do it however you like. Different forms make sense to different people. If we're not mistaken, you're a very different person.