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### Course: Algebra 2 (Eureka Math/EngageNY) > Unit 1

Lesson 13: Topic C: Lesson 32: Graphing systems of equations- Interpreting equations graphically
- Interpreting equations graphically (example 2)
- Interpret equations graphically
- Solving equations graphically (1 of 2)
- Solving equations graphically (2 of 2)
- Solving equations graphically
- Solve equations graphically

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# Solving equations graphically (2 of 2)

Sal solves the equation e^x=1/[x(x-1)(x-2)] by considering the graphs of y=e^x and y=1/[x(x-1)(x-2)]. Created by Sal Khan.

## Want to join the conversation?

- Where can i get this calculator that sal uses?(10 votes)
- Here's a website with a list of emulators:

http://www.ticalc.org/basics/calculators/ti-85.html#10

Hope that helps. :D(14 votes)

- If we're trying to find where
`E(x)`

and`R(x)`

are the same, why not graph`E(x)-R(x)`

?(9 votes)- Yes, that would work. If you're trying to find where E(x) = R(x) then you can alternatively try to find where E(x) - R(x) = 0. The result will be the same.(12 votes)

- What does it mean to be within 0.01?(4 votes)
- That the difference between the two is less than or equal to 0.01. So |E(x)-R(x)| <= 0.1(2 votes)

- As far as learning the calculator better...other than the Texas Instruments site itself, are there any YouTube videos etc. that could be recommended?(2 votes)
- Why doesn't e equal 10 because it is suppose to be the base 10?(1 vote)
- No. e is the base of the natural logarithm.

http://en.wikipedia.org/wiki/Natural_logarithm

You are reffering to the base 10 logarithm or common logarithm.

http://en.wikipedia.org/wiki/Common_logarithm

They are two very different logarithmic operations.

http://en.wikipedia.org/wiki/Logarithm(2 votes)

- In the exercises for this concept, how do I know which value is X1 and X2?(1 vote)
- What is the best calculator that has graphs, trig functions, e^x etc?(1 vote)
- Would this kind of calculator be allowed on SAT exam?(1 vote)
- Is it possible to solve log_2(x+4) = 3 -x ?(1 vote)
- wait, why are some graphs curved? I thought all graphs had to be straight...(1 vote)
- The only graphs with straight lines are the graphs of linear functions. Everything with a degree greater than 1 will have curves. The only rule on graphs of function is the vertical line test. If there does not exist a vertical line that passes through no more than 1 point on the curve then that graph is the graph of a function. In fact you'll eventually learn that a line is a special type of curve just like a square is a special type of rectangle.(1 vote)

## Video transcript

In the last video, we estimated
the solution to e to the x is equal to 1 over x times
x minus 1 times x minus 2 using a calculator. We got a first rough estimate
by just looking at this graph, and then we tried values out to
truly zero in on, or get close to the x value
where this is true. What I now want to
do is actually just use the graphing functionality
of this calculator to try to estimate the
solution graphically. So let's go to Graph, and
what I'm going to try to do is graph both of
these functions. So the first
function-- let me clear this-- the first
one is e of x, which on the graphing
calculator will be y1. And that's going to
be e to the x power, and then the second
one, y2, will be r of x, which is
going to be 1 divided by x times x minus
1 times x minus 2. And so, let's see. I have to close this
parentheses as well. So I've entered it in the graph. And I care about
where we zoom in, so I really want to zoom
into this part of the graph right over here. So let me go to the Range. So, actually there's also a Zoom
functionality that I could use, but let's-- actually,
let me do that. That could be fun, so
let's just graph it. Actually, let's
just see what range it's graphed at right now. Let's see. What we would care
about, let's start with a rough
approximation, just to see that this is indeed
the same graph. So let's start with x
going from 0 up to-- I don't know-- 3, so this would
be this part of the graph right over here. And then the x scale is 1. That's what they'll
mark off, every 1. We could even mark off
every 0.5 if we want. Like this one is
marked off every 0.5. And the y minimum, let's go
from 0 to-- on this range, actually it goes pretty high,
and the way this is graphed, it goes all the way up to,
looks like 10, so I'll go 10. I'll leave the y scale as 1. They mark it off every
1 right over here. And now let's graph this thing. And that was e of x, and
now it's graphing r of x. And you see it
indeed looks pretty similar to what we have here. Now what we care about is this
point, or on our calculator, this point right over here. We want to figure
out what x value-- what is the x-coordinate of
this point of intersection? This is when our two functions
are equal to each other. So let me zoom in on this. I think I can use this
Box functionality. So it essentially lets me
construct a box around this, and it's going to
zoom into that box. So I'm going to get as tight
in on this as I can go. So if I press, I can
get even tighter on it. So if I press Enter, now I
can define the other corner of the box, so
that's pretty good. I'm going to zoom
in, press Enter, and that's zoomed in on
to that little teeny box. So that was e of x, and now
it's going to graph r of x. So now let me try
to trace the graph. So let's see. Trace, so it's letting
me trace e of x. And let's see. If I look at my x values
decrease-- so at this point, e of x is still
higher than r of x. And if we get right over here--
so 2.056, we see that r of x is above e of x. We just see that
graphically, and then we're left of the point
of intersection. And then we're still left of
the point of intersection. Now we're right of the
point of intersection, so it looks like the
point of intersection is between 2.057 and 2.059. And so in the
previous video when we said where our
estimate was 2.06, we were definitely within 0.01
of the point of intersection. If we did want to get even more
precise, we could zoom in more, and I encourage you if you've
got a graphing calculator like this to actually
try that out.