Two-Variable Equations - Online Tutor, Practice Problems & Exam Prep

Equations can involve one or two variables, with the latter represented as ordered pairs (x, y) on a two-dimensional plane. For example, the equation $x+y=5$ has infinite solutions. To graph such equations, isolate y and calculate corresponding x values, plotting points to form a line. Additionally, intercepts are points where graphs cross the axes, with x-intercepts having a y-value of 0 and y-intercepts having an x-value of 0, crucial for understanding graph behavior.

Equations with Two Variables

Hey, everyone. So up until now when we've solved equations, they've always been of one variable, like x + 2 = 5. But equations in this course won't always be so simple. Instead of just one variable, a lot of equations in this course will now start to involve two variables, and the most common ones you'll see are going to be x and y. So, for example, instead of x + 2 = 5, now we're going to have x + y = 5. I'm going to show you the main difference between these two types of equations and how to solve them and, more importantly, how to visualize them. So let's go ahead and get started here.

With equations with one variable, like for example, x + 2 = 5, you're really just trying to find the value of x that makes this equation true. So you would isolate and solve for x. You would subtract 2 from both sides, and your answer ends up being 3. That's the number that makes this equation true. We visualize it by plotting it on a one-dimensional number line, like this point over here. Now, let's take a look at x + y = 5. Now we actually have two variables, x and y. Both are letters that can be replaced with numbers. So, how do we solve this? Well, one thing you might notice is that if you try to isolate one of the variables, it's not really going to be much help because x = 5 - y doesn't give you any information about what x or y could be.

Let's think about what this equation means. Can I think of two numbers that, when I add them together, I get 5? And, actually, yes. I can. Because, for example, if x = 1, what do I have to add to 1 to get to 5? Well, y could be 4. So, that's a solution to this equation. Both of these numbers here, this combination makes this equation true. But is this the only combination of numbers that makes this true? Well, actually, no. Because what if x = 2? If x = 2, then y could be 3, and that also satisfies this equation, x + y = 5. In fact, you could have another one, x = 5, and then y could just equal 0. So the whole point here is that with equations with one variable, the solution was always just one number. It was a single point on a one-dimensional plane versus when you have equations with two variables, as we can see, you end up with many solutions that satisfy this equation. And the way that we represent them is not on a number line, but actually as points as ordered pairs x,y on a two-dimensional plane.

So, basically, what happens is I can take x = 1, y = 4 and turn it into an ordered pair 1,4. This becomes 2,3, and this becomes 5,0. And we can plot these on a two-dimensional plane. So, for example, 1,4, this point satisfies this equation. 2,3 also satisfies this equation. And 5,0 also satisfies this equation. So all these things actually satisfy. In fact, there's actually an infinite number of solutions because if you sort of keep this pattern going on here, any points that are basically on this line over here will actually satisfy between these two types of equations.

Now, how do we actually use this in problems? Well, let's take a look at our example here. In our example, we have this equation, x + y = 5. This is the same exact equation we've been using before, and we want to first determine if these points satisfy the equation. We've got 3, 2, 4, 1, 0, 0, and -1, 3. So, if you're ever asked to determine whether points x,y satisfy an equation, what they're really just asking you to do is they're asking you to replace the x and y values to check if the equation is true. Remember, that's what satisfying an equation means. So what we're doing in part a is when we have a coordinate like 3, 2, this is really just giving us an x and a y value, and we just plug it into this equation, x + y = 5, and we just figure out if that makes the equation true. So does it? Well, if I basically replace x with 3 and y with 2, then 3 + 2 does, in fact, give me 5. So this definitely does satisfy the equation. Let's get started with the next one, which is 4, 1. Again, does this satisfy x + y = 5? Well, if I replace x with 4 and y with 1, this does indeed get me 5. So both of these points do satisfy the equation. What about 0, 0? I do 0, 0 into x + y = 5, then I'm just replacing x with 0, y with 0, and 0 + 0 does not give me 5. So this actually does not turn out to be a solution to this equation. And last but not least, what about -1, 3? So into x + y = 5 this is going to be -1 + 3, and this also does not give me 5. It gives me 2. So, it turns out that not all of these points are going to satisfy the equation. The first two work. The second two didn't work. And now we're going to go ahead and plot them.

Right? So, as we can see, the point 3, 2 is going to be here. That's the point 3, 2. Right? That's this point over here. We've got 4, 1. This is also going to be a solution to the equation. And then we've got 0, 0, which is over here, and then -1, 3. So if you've seen here, there's a pattern that happens. Basically, when points do satisfy an equation, when they do make the equation true, then they are on the graph of that equation versus when they do not satisfy an equation, as we've seen with these points over here, they are not on the graph of that equation. So that's the basic difference between equations with two variables versus one variable. Alright? So, hopefully, that made sense. Thanks for watching.

Graphing Equations of Two Variables by Plotting Points

Hey, everyone. So up until now, when we've seen equations with 2 variables, their graphs were already given to us. For example, we saw that x+y=5 was a line that looks like this. But sometimes their graphs aren't going to be given to us, and you're going to have to go graph them yourself. That's what I want to show you how to do in this video, and, basically, the way we're going to do this is by plotting points. To graph an equation, we're going to calculate and plot a bunch of ordered pairs that make the equation true, and the way we do that is by plugging a bunch of x values into this equation, and we're going to get points out of it. So those points we can take and plot them on a graph and then connect them with a line. That's basically what I'm going to show you how to do in a step-by-step way. Let's get started. Alright. So we're just going to actually dive right into our equation or our example here so I can show you how this works. So we're going to graph this equation -2x+y=-1, and we're going to do this by creating a bunch of ordered pairs using some x values that are already chosen for us. So here's the first step. You're going to take an equation, and if it's not already isolated to one side, you're going to have to isolate y to the left side. So, basically, you always want y equals and then the rest of the equation on the right side. And the reason to do this is because you're going to plug x values into this equation, and it's just going to be way easier to see what y is if it's isolated. Alright? So how do we do this? How do I take this equation over here and isolate y? Well, basically, all I have to do is just take the negative two x, and move it to the other side. So I'm going to add 2x over here and add 2x to both sides. And, basically, what I end up with is when I bring it down here, this is actually just going to end up being 2x-1. So now that y is isolated, I can move on to the next step. What I want to do is I want to calculate y values for a bunch of x values that I choose, and usually 3 to 5 is enough. So I've got my table here that we've got a bunch of x values that are already chosen for me, negative 2 and so on. And what I'm going to do here is I'm just going to replace the x inside of this equation with whatever that x value is, and then I'm basically just going to get a y value out of it. So for example, I'm just going to take this negative 2, and I'm going to plug it in. Right? So this is going to be 2 times negative 2 and then minus 1. So what does that give me? What y value does that give me? Well, 2 times negative 2 is negative 4, and negative 4 minus 1 is actually just going to give me negative 5. So this is going to be negative 5 over here. What that means here is that I plugged in negative 2, and I got out of it negative 5. So that's an ordered pair. The ordered pair that satisfies this equation is negative 2 comma negative 5. You're just going to repeat this a bunch of times over here. So this y equals 2 times negative one minus 1. So this is going to be negative 2, minus 1, which is going to be negative 3. So this is going to be negative 2 comma negative 3. 0 is always a pretty easy point to do because usually what happens is you just replace x with 0 and one term just kind of goes away. So this whole term just cancels out, and then you get the negative one, which is just negative one. So the ordered pair is 0 comma negative one. And then over here, we're going to get 2 times 1 minus 1, so that's 2 minus 1, and that's just going to give us positive 1. So this is going to be 1 comma 1. And then finally, we've got 2 times 2, which is 4, and then 4 minus 1 is equal to 3. So the ordered pair is 2 comma 3. Alright? So you just have to basically just plug and chug a bunch and fill out your table. Tables are going to be really good at organizing your information. And the last thing you're going to do, or sorry. The third thing you're going to do is you're going to plot these xy points that you've just gotten in this table over here, and you're going to plot them on your graph. So let's go ahead and do that. So I'm going to take the ordered pair negative 2 comma negative 5, and that's going to be right over here. I'm going to take the points. I'm sorry. This is going to be well, whoops. That's going to be negative 1. I'm sorry. Hopefully, you caught that. So negative one comma negative three is going to be this point over here, then we've got 0 comma negative one. So that's over here. And we've got 1 comma positive 1, which is over here. And then finally, we got 2 comma 3 over here. So I just end up with a bunch of points, I'm going to plot them all, and the last thing I do is connect them with a line or sometimes a smooth curve. In this case, what happens is I can actually draw a line that goes through all of these 5 points, and so this is the graph of the line. So this is the graph of the equation, y equals 2x minus 1 or negative 2x plus y equals negative 1. That's how to graph these types of equations. The very last thing I want to sort of mention here is that most of the time these x coordinates won't be given to you, and if you're not given x values to evaluate, then you can always just feel free to choose your own. And usually, 3 to 5 is enough. Alright? So thanks for watching. That's it for this one.

Graph the equation $y-x^2+3=0$ by choosing points that satisfy the equation.

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Graph the equation $y=\sqrt{x}+1$ by choosing points that satisfy the equation. (Hint: Choose positive numbers only)

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Graphing Intercepts

Hey, everyone. So throughout some of your graphing problems, you may be asked to identify these things called intercepts, and that's what I want to show you how to do in this video. Basically, what we're going to see here is that intercepts are really just special places or special points where the graph crosses either one of the x or y axis. So I'm going to show you how to identify those types of points and also some special things about them, and we'll do an example. Let's get started.

Alright? So in this equation, I've got or in this diagram here, I've got 2 lines. I'm going to call this one a and this one b. Both of these graphs actually cross the x and y axis. If you look at line a, line a crosses the x axis over here. Line b crosses the x axis over here. These things are called the x intercepts because that's where they cross the x axis. So the x intercept is where the graph crosses the x axis. Now, similarly, these graphs also cross the y axis over here and over here. So these points over here are called the y intercepts. That is basically the y value where the graph crosses the y axis.

Alright? So pretty straightforward here. If you take a look at line a, line a crosses over here. So this is an x intercept, where x is equal to negative 2. And over here, this is where x is equal to negative 4. So it's basically just the x value where that graph crosses the x axis. Alright? And over here, what we've got is we got y equals 4, and over here, we've got y equals negative 3. Alright? So that's what the y intercepts and x intercepts are. Now what's sort of sort of confusing about this is what happens to the other value. So, for example, in the x intercepts, what you're going to see here is that the y value at any point along this line is always equal to 0 because you're basically right on the line, you have no height above it. So the x intercept is where across the x axis, but it's also where the y value is equal to 0. Now similarly, what happens for the y intercept is that the x value is going to be 0. So notice how here, anytime you're on the y axis, you're not to the right or to the left of the origin. So the y intercept is where it crosses the y axis, but the x value is always 0. That's kind of like the opposite. Alright? That's really all there is to know about x and y intercepts. So let's go ahead and do a quick example.

So here, we're going to take this graph, and we're going to write the x and y intercepts of the graph that's shown below here. So remember, the x intercepts were crosses the x axis. If you look at this line or this sort of curve shaped over here, this curve crosses the x axis twice, once over here and what's over here. So what happens here is that the x intercepts are just the values. It's just the number of the x coordinate. So in other words, our x intercepts are negative 3 and positive 5. You don't have to write the ordered pair. You just have to write the numbers. Then what happens for the y intercept? Well, this graph also crosses the y axis over here. That's going to be a y intercept. So your y intercept over here is going to be y equals negative 4. So that's really all there is to it. Right?

So let's take a look at another example here. We're now the question is written a little bit differently because instead of having to write the x intercepts and y intercepts, we're just asked to find the intercepts of the graph below. Okay? So let's take a look at that. So in this graph here, we've got this sort of like circle looking thing, and notice how this graph actually crosses the x axis right over here. So basically, what happens here is when they're asking for the intercepts, they're actually not asking for the x value. They're asking for the ordered pair. So that's sort of the real confusing thing with these sort of problems. But, basically, if they ever ask for an x or y intercept whenever they actually reference the letter by name, then all you're actually have to do is just write the x or y value. But if they just ask you to find the intercepts like they have like they're asking you for in the second example, then you're going to write the ordered pair. So, really, all we have to do for the second example is just write the ordered pairs for these two numbers. And in this case, it's just going to be (2, 0), and then also, and then it's going to be (4, 0). So these are the intercepts of the graph, but these are the x and y intercepts of this graph. Alright. So that's really all there is to intercepts. Thanks for watching.

Here’s what students ask on this topic:

What is the difference between one-variable and two-variable equations?

One-variable equations involve a single variable, such as $x+2=5$. The solution is a single value that makes the equation true, represented on a one-dimensional number line. Two-variable equations, like $x+y=5$, involve two variables and have infinite solutions. These solutions are represented as ordered pairs (x, y) on a two-dimensional plane. The main difference is that one-variable equations have a single solution, while two-variable equations have multiple solutions that form a line or curve on a graph.

How do you graph a two-variable equation?

To graph a two-variable equation, follow these steps: 1) Isolate y on one side of the equation, so it is in the form $y=\mathrm{mx}+b$. 2) Choose several x-values and calculate the corresponding y-values. 3) Create ordered pairs (x, y) from these values. 4) Plot these points on a two-dimensional graph. 5) Connect the points with a line or curve. For example, for the equation $-2x+y=-1$, isolate y to get $y=2x-1$, then plot points like (-2, -5), (-1, -3), (0, -1), (1, 1), and (2, 3).

What are intercepts in a graph, and how do you find them?

Intercepts are points where a graph crosses the x-axis or y-axis. The x-intercept is where the graph crosses the x-axis, and the y-value is 0. The y-intercept is where the graph crosses the y-axis, and the x-value is 0. To find the x-intercept, set y to 0 in the equation and solve for x. To find the y-intercept, set x to 0 and solve for y. For example, in the equation $y=2x-1$, the y-intercept is (0, -1) and the x-intercept is (0.5, 0).

How do you determine if a point satisfies a two-variable equation?

To determine if a point (x, y) satisfies a two-variable equation, substitute the x and y values into the equation and check if the equation holds true. For example, for the equation $x+y=5$, to check if the point (3, 2) satisfies it, substitute x = 3 and y = 2 into the equation: $3+2=5$. Since the equation is true, the point (3, 2) satisfies the equation.

What is the significance of ordered pairs in two-variable equations?

Ordered pairs (x, y) are significant in two-variable equations because they represent the solutions to the equation on a two-dimensional plane. Each ordered pair is a point that satisfies the equation. For example, in the equation $x+y=5$, the ordered pairs (1, 4), (2, 3), and (5, 0) are all solutions. These points can be plotted on a graph to visualize the relationship between x and y, forming a line or curve that represents all possible solutions.