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### Course: Algebra basics > Unit 4

Lesson 2: Solutions to two-variable linear equations# Two-variable linear equations intro

Learn about a class of equations in two variables that's called "linear equations." They are called that way because their graph is a line. These are the most basic and probably most useful equations you will ever know!

## Want to join the conversation?

- If x is an irregular number such as pi, can the equation still be linear? Thanks!!(34 votes)
- Yes, it's just harder to graph, as the graph uses increments of whole numbers for the most part, but just imagine a point that's about 3.14 or some other point on the graph that's very close to the result you are looking for.(41 votes)

- I have come from bangladesh to u.s.a and now iam will go to high school 10 grade,, can u tell me what type of math i have to peactice(19 votes)
- In the US, 10th grade math is usually either Geometry or Algebra II and/or Trig. You probably want to make sure you know everything in Algebra 1 (8th or 9th grade) (https://www.khanacademy.org/math/algebra-home/algebra-basics) and (https://www.khanacademy.org/math/algebra-home/algebra). If you are doing algebra II (depending on the school) you may want to review geometry (https://www.khanacademy.org/math/geometry-home). Or you may want to look over algebra II (https://www.khanacademy.org/math/algebra-home/algebra2). Depending on what your school was like in Bangladesh, you may be used to this way of grouping classes, or you may be learning some geometry and some algebra each grade.(38 votes)

- So if I just plug a number in the x value, it will lead me to the y value? And then I can just graph it and it will be in line with the other points?(10 votes)
- Find the mid point of line joining the point (22,20) and (0,16)(14 votes)

- what is simutaneos equation(6 votes)
- A Simultaneous Equation is a finite set of equations which has common solutions. It is same as Systems of linear equations.(19 votes)

- Is a linear equation the same thing as a proportional relationship? Don't proportional relationships have to pass through the origin though?(6 votes)
- Yes, a proportional relationship is a linear relationship that goes through the origin, so not all linear relationships are proportional.(10 votes)

- Why does the solution of a linear equation always come out to be a straight line on the graph?(4 votes)
- Good question!

In x and/or y, any linear equation is equivalent to one of two forms: x=a or y=mx+b where a, m, and b are constants. (Yes, this already includes the form where y is a constant, because this would be the result of taking m to be 0 in the equation y=mx+b).

For the form x=a, the graph is the set of all points with constant x-coordinate a. This is clearly a vertical line through (a, 0).

For the form y=mx+b, if we can show that the slope is constant, then the graph must be a straight line. For any two distinct points on the graph, the x-coordinates, say x_1 and x_2, are also distinct (since the same value of x would give the same value of y because y is given explicitly in terms of x). The two points are (x_1, mx_1+b) and (x_2, mx_2+b). The distinctness of x_1 and x_2 avoids division by 0 when we find the slope.

For these two points, the slope is

[(mx_2+b)-(mx_1+b)]/(x_2-x_1)

= (mx_2-mx_1)/(x_2-x_1)

= m(x_2-x_1)/(x_2-x_1)

= m.

The slope is the constant m, so the graph of y=mx+b is a straight line.

We conclude that the graph of any linear equation in x and/or y is always a straight line.(13 votes)

- what is the point in this its not like we are going to need to know this stuff to get into the army?!(1 vote)
- Well, not everybody is in the army and that isn't the only thing you can do in life. You're going to need to know this stuff in order to help advance the world's technology.(10 votes)

- What would a linear equation look like in the First, Third, or even Fourth dimensions?(6 votes)
- A line will still be a line in whatever dimensions you are talking about.(2 votes)

- Is there a function like this:

x=yz

or xa=yz?

And what's the difference between linear equations and functions?(5 votes)- The way my teacher writes functions is O(I)=R

O = the output usually y

I = The input variable usually x

R = The functions rule to compute a situation(2 votes)

- So if you put any number for an equation, it will still form a line no matter what? :3(3 votes)
- Not exactly sure what you mean, it really is the other way around, if you have a linear function (which forms a line), then you can input any value in and you would get the accompanying output value that is on the line. So basically you start with the fact that it is a line, then you can find different points that are on the line. Is this what you are asking, or are you asking something else?(3 votes)

## Video transcript

- [Voiceover] What I'd like to
introduce you to in this video is the idea of a Linear Equation. And just to start ourselves out, let's look at some examples
of linear equations. So, for example the equation y is equal to two x minus three, this is a linear equation. Now why do we call it a linear equation? Well if you were to take
the set of all of the xy pairs that satisfy this equation and if you were to graph
them on the coordinate plane, you would actually get a line. That's why it's called a linear equation. And let's actually feel
good about that statement. Let's see, let's plot some of the xy pairs that satisfy this equation
and then feel good that it does indeed generate a line. So, I'm just gonna pick some x values and make it easy to calculate
the corresponding y values. So, if x is equal to zero y is gonna be two times zero minus three which is negative three. And on our coordinate plane here that's-- we're gonna move zero in the x direction, zero in the horizontal direction and we're gonna go down three
in the vertical direction, in the y direction. So, that's that point there. If x is equal to one, what is y equal to? Well two times one is two,
minus three is negative one. So we move positive one in the x direction and negative one, or down
one, in the y direction. Now let's see, if x is
equal to two what is y? Two times two is four, minus three is one. When x is equal to two y is equal to one. And hopefully you're seeing now that if I were to keep going, and I encourage you though if you want pause the video and try x equals three or x equals negative one and keep going. You will see that this is
going to generate a line. And in fact, let me connect these dots and you will see the line
that I'm talking about. So, let me see if I can draw, I'm gonna use the line tool here. Try to connect the dots
as neatly as I can. There you go. This line that I have just drawn, this is the graph, this is the graph of y is
equal to two x minus three. So if you were to graph all
of the xy pairs that satisfy this equation you are gonna get this line. And you might be saying,
"Hey wait wait, hold on Sal, you just tried some particular points, why don't I just get a bunch of points, how do I actually get a line?" Well, I just tried, over here I just tried
integer values of x. But you can try any value in
between here, all of these, it's actually a pretty unique concept. Any value of x that you input into this, you find the corresponding value for y, it will sit on this line. So for example, for example, if we were to take x is equal to, actually let's say x is
equal to negative point five. So if x is equal to negative point five if we look at the line when x is equal to negative point five it looks like it looks like y is equal to negative four. And that looks like that sits on the line. Well let's verify that. If x is equal to negative, I'll write that as negative one half, then what is y equal to? Let's see, two times negative one half, I'll just write it out, two times negative-- two times negative one half minus three. Well this says two times
negative one half is negative one minus three
is indeed negative four it is indeed negative four. So you can literally take any, any-- for any x value that you put here and the corresponding y value it is going to sit on the line. This point right over
here represents a solution to this linear equation. Let me do this in a color you can see. So this point represents a
solution to a linear equation. This point represents a
solution to a linear equation. This point is not a solution
to a linear equation. So if ex is equal to five then y is not gonna be equal to three. If x is gonna be equal to
five you go to the line to see what the solution to
the linear equation is. If x is five this shows us
that y is going to be seven. And it's indeed-- that's indeed the case. Two times five is ten,
minus three is seven. The point-- the point five comma seven is on, or it satisfies this linear equation. So if you take all of the
xy pairs that satisfy it, you get a line. That is why it is called
a linear equation. Now, this isn't the only
way that we could write a linear equation. You could write a linear equation like-- let me do this in a new color. You could write a linear
equation like this: Four x minus three y is equal to twelve. This also is a linear equation. And we can see that if we
were to graph the xy pairs that satisfy this we would
once again get a line. X and y. If x is equal to zero, then this goes away and you have negative three
y is equal to twelve. Let's see, if negative
three y equals twelve then y would be equal to negative four. Nega-- zero comma negative four. You can verify that. Four times zero minus
three times negative four well that's gonna be
equal to positive twelve. And let's see, if y were to equal zero, if y were to equal zero then this is gonna be four
times x is equal to twelve, well then x is equal to three. And so you have the point
zero comma negative four, zero comma negative four on this line, and you have the point three
comma zero on this line. Three comma zero. Did I do that right? Yep. So zero comma negative four
and then three comma zero. These are going to be on this line. Three comma zero is also on this line. So this is, this line is
going to look something like-- something like, I'll
just try to hand draw it. Something like that. So once again, all of the xy-- all of the xy pairs that satisfy this, if you were to plot them
out it forms a line. Now what are some examples,
maybe you're saying "Wait, wait, wait, isn't any
equation a linear equation?" And the simple answer is
"No, not any equation is a linear equation." I'll give you some examples
of non-linear equations. So a non-- non-linear, whoops let me write a
little bit neater than that. Non-linear equations. Well, those could include something like y is equal to x-squared. If you graph this you will see that this
is going to be a curve. it could be something like x
times y is equal to twelve. This is also not going to be a line. Or it could be something
like five over x plus y is equal to ten. This also is not going to be a line. So now, and at some point you could-- I encourage you to try
to graph these things, they're actually quite interesting. But given that we've now seen
examples of linear equations and non-linear equations, let's see if we can come
up with a definition for linear equations. One way to think about is it's an equation that if you were to graph
all of the x and y pairs that satisfy this equation, you'll get a line. And that's actually
literally where the word linear equation comes from. But another way to think about it is it's going to be an equation where every term is either
going to be a constant, so for example, twelve is a constant. It's not going to change
based on the value of some variable, twelve is twelve. Or negative three is negative three. So every term is either
going to be a constant or it's going to be a
constant times a variable raised to the first power. So this is the constant two
times x to the first power. This is the variable y
raised to the first power. You could say that bceause
this is just one y. We're not dividing by x or y, we're not multiplying, we don't have a term that
has x to the second power, or x to the third power,
or y to the fifth power. We just have y to the first power, we have x to the first power. We're not multiplying x and y together like we did over here. So if every-- if every term in your equation, on either side of the equation, is either a constant or its
just some number times x, just x to the first power
or some number times y, and you're not multiplying
your x's and y's together you are dealing with a linear equation.