When it comes to searching for buried treasure,
any child who’s dreamt of being a pirate or great adventurer can tell you: X marks
the spot.
Later in life, sacks of doubloons and rubies are a little harder to come by,
but we’re still on the hunt -- this time for answers.
Today we’ve traded in our treasure maps for graphs.
There’s a wealth of information hidden where two equations have the same solutions,
and graphing and seeing where two curves intersect can guide us there faster.
Now some of these intersecting questions show up where you might expect,
like in high-stakes decisions made by people like economists, contractors, or air traffic controllers.
But they’re also woven through the fabric of everyday life in the form of criss-crossing
linear equations
-- or equations that describe a relationship between two variables where we can multiply
one variable by a constant to get the other.
We call them linear because they form straight lines when we graph them.
Linear equations, like a compass to our fearless swashbucklers, are essential to algebra
and describe fundamental relationships like the one between distance, speed, and time,
or price and cost.
So let's see what clues we can dig up and how we can solve equations just by graphing.
G’Day, I’m James Tanton, and this is Study Hall Algebra presented by Arizona State University
and Crash Course.
Linear equations show up all over algebra, so it’s easy to feel like you either already
know everything about them,
or like you should already know everything about them.
But neither of those things have to be true.
Like, here’s a surprisingly hard question to answer: what makes a straight line straight?
We can close our eyes and imagine a straight line, or just draw one with a ruler.
But how do we describe that picture in the language of math?
We can start by looking for patterns that are unique to graphs of lines.
For instance, look at the angle between the line and any other horizontal segment.
No matter where you are along that line, the angle will always be the same.
Compare that to a graph of a quadratic equation. The angle keeps changing!
Or we might notice how the line moves across the plane.
If I were to run a fixed distance to the right, no matter where I started,
I would have to rise the same amount each time to get back to the line.
And if someone doubled my run, they’d have to double my rise!
So to answer our not-so-obvious question: a line is straight if the ratio of rise over run
-- which we call the slope -- is the same number, no matter where you choose to measure
that ratio along the line.
You can also think of the slope as measuring steepness.
This line has a sharp slope of 4; this line has a gentle slope of ½.
If the rises and runs we draw look like an “up” staircase from left to right,
we have a positive slope. If they look like a “down” staircase, then the slope is
negative.
Textbooks give this formula to find the slope, but there’s no need to memorize it.
After you pick any two points on your line, the bottom part tells you how far you’ve
run and the top part tells you how far to rise.
And the line is straight if we get the same ratio no matter which two points we pick.
Now there are two types of lines that cause trouble in the slope department.
One is a perfectly flat horizontal line like this one.
This line represents all the points that make the equation y = 3 true.
So no matter where I start, if I run some distance to the right, how far do I need to
rise or fall to get back to the line?
Zero! I never left. So the slope of a horizontal line is zero.
The second challenge is in the other direction.
Start at any point on this vertical line and run any distance to the right.
Now try to rise or fall back to the line.
We’ll never get there! Vertical lines break all our rules for slope, so we call the slope
“undefined.”
But since this is a valid line, the language of math is still able to talk about it.
In this case, this line represents all the points that make the equation x = 5 true.
Now if you’ve taken an algebra class before and someone leapt out of a crowd,
demanding you tell them about graphing linear equations, out of sheer muscle memory you
might say “Uh … y equals mx plus b!”
But the person next to you might remember something about getting the slope from two points.
And another person might remember an equation with A and B and C.
From the sound of it, we’ve got a whole scatterplot of knowledge.
But in fact, we’re all talking about the same thing.
It turns out there are lots of ways to write linear equations with math, and they’re
all useful in different ways.
But I’ll be straight with you -- this is just one of those things we have to learn,
like different tenses while learning a language.
Math is a language too, so with practice you'll learn to recognize these forms.
Now in a textbook we might see something like:
“Find the equation for a line with a slope of 2 that passes through the point (3, 4).”
We could put those pieces into our slope equation.
For any point (x,y) on the line, the rise over the run must equal 2 over 1, and we can
plug in (3,4) for one of the points in the formula.
Then (y – 4) over (x – 3) equals 2.
Now this is a good start for an equation, but it needs some adjustment.
The way it is now, it has a fatal flaw: we’re told (3,4) is on the line, but if we plug
in x = 3, what happens?
GASP! We’d be dividing by 0!
We can get around this flaw by multiplying both sides by (x - 3) and use the equivalent
equation y-4 = 2(x-3) instead.
Now even the point x=3, y=4 gives a true sentence.
Crisis averted!
Because we built that equation up from just a single point and the slope,
that general strategy is called finding the point-slope form.
But thanks to how graphing calculators work and because we tend to like pictures,
you might be more familiar with the slope-intercept form.
In the slope-intercept form, some key parts of the graph are easy to visualize from the
equation, like the slope and the y-intercept
-- or the point where the line crosses the vertical axis.
Looking at our horizontal axis, we can see the y-intercept is always where x = 0,
and that's how we can solve for it.
If we already have the point-slope form,
we can get the slope-intercept form pretty easily by putting everything except the y
on one side.
The slope of the line is 2, and if we plug in x = 0, we get the y-intercept -- negative 2.
There’s one more linear equation form, standard form.
Here we rearrange the equation so we get all the variables on one side and all the constants
on the other.
Standard form is useful because it doesn’t focus on the slope, so we can still use it
to describe those pesky vertical lines.
The line x = 5 is really 1x + 0y = 5. It was in standard form all along!
Linear equations can come in a few different forms, and no form is inherently better than
any other.
You can write it as y = mx +b or Ax + By = C, or some other clever way that you like.
Mathematics cares more about representing the truth than about convention.
And you might switch it up depending on the situation.
For comparing two lines, we might choose a form where the slope is clear because it hints
at how the lines behave together.
For instance, we can think of two lines with the same slope as “moving at the same pace.”
If I start on a line and run one unit to the right and then rise 2 units to get back to
the line,
I’ll have to do the same thing on the second line.
The two lines will never intersect and are parallel.
Perpendicular lines are sort of the opposite and cross at crisp right angles.
Here’s a line with a rise of 3 and a run of 2.
Rotate the picture 90 degrees and we get a perpendicular line. Now it has a run of 3
and rise of -2.
In general, perpendicular lines have slopes with opposite signs that are reciprocals of
each other
-- meaning if we know the slope of one line, we flip the fraction and swap the sign to
get the slope of the second.
So if we have two or more equations in point-slope or slope-intercept form, we’ll be able to
tell if the lines are parallel or perpendicular before we’ve done any graphing!
But what about lines that aren’t parallel or perpendicular?
If they’re not parallel, they must intersect somewhere -- the trick is figuring out where.
In fact, hunting for the place where X-marks the spot -- or where two curves intersect
-- is a big deal in math,
not just with linear equations.
Instead of chests of gold, the spot where two curves meet is a treasure trove of information.
Whatever relationships we’re graphing, they only have the same values where the lines cross!
Finding these intersections is actually solving simultaneous equations, which are sets of
two or more equations that all use the same variables.
We’re looking for the values that make all those equations true at the same time.
If we have to solve by hand, this is standard form’s chance to shine.
But we can usually find the answer just by graphing instead.
Let’s start our treasure hunt with a word problem.
“How many gallons of 50% salt solution and how many gallons of 10% salt solution
would have to be combined to make 4 gallons of 20% salt solution?”
Wait! This is word problem déjà vu.
We solved something like this with pictures! Now we can try with graphs.
Let’s focus on this through X, Y, and Grid-Ray Vision.
We know we need to end up with 4 gallons of solution that’s 20% salt -- or
has 4 times .2 = .8 gallons of salt.
And to graph, we’ll need to turn our information into equations.
I’ll call the amount of 10% solution we have K for Kinda Salty and the amount of 50%
solution V for Very Salty.
Then K and V added together need to sum to 4.
Now we need a second equation to bring in the salt.
The amount of salt from the Kinda Salty solution will be 0.1 times K
-- because 10% is salt -- and the amount of salt from the Very Salty solution will be
0.5 times V -- because 50% is salt.
Added together that’s our total amount of salt.
We’re more likely to make mistakes with decimals,
so let's multiply both sides of this equation by 10 and get an equivalent equation.
To start plotting, let’s put V horizontally and K vertically.
If we find the V and K-intercepts, we can use those two points to sketch the line.
That’s easy for the first equation.
And the K-intercept for the second one is pretty straightforward: when V = 0, K = 8.
But what about the K-intercept?
When K = 0, V = 8/5. That could work, but integer data points are simpler.
Instead, we notice when V is 2, then K has to equal negative 2.
Does that make any sense in the real world? Nope.
We can’t have negative volume. But it’s fine to extend our math to help us out.
And the drumroll moment. Do … they … intersect!
They do, and our answer sounds familiar from our last salt scenario.
One gallon of 50% salt plus 3 gallons of 10% salt equals 4 gallons of 20% salt solution.
So for the second time in a row, we’ve solved a solution-dilution problem just by drawing
a picture of its relationships.
But this time, we did it a little more formally.
And we could use this graph to get other information about the situation, like if we wanted to
adjust the ratios.
Now if you are asked to find the equation of a line -- even buried in a word problem
-- here are some starting points.
The slope formula is just finding the rise over the run between two points.
Use it to find an equation for your line, whether you already have one point and the
slope or just two points.
And if you’re solving a system of linear equations, graph them and see where they intersect.
A quick way to get a couple data points to work with is to set each variable to zero.
Next time we’ll look at the graphs of other equations we haven’t drawn yet, like logarithms,
and see what else we can learn from graphs to help us in the real world.
Until then, cheers!
Thanks for watching Study Hall Algebra, which is produced by Arizona State University and
the Crash Course team at Complexly.
If you liked this video and want to keep learning with us here in Study Hall, be sure to subscribe.
You can learn more about ASU and the videos produced by Crash Course
in the links in the description.
See you next time!