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Solving a System of Linear Equations using the Addition Method

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Hi, my name is Anthony Varela. And in this tutorial, we're going to be solving a system of linear equations algebraically using what we call the addition method. So the addition method is all about adding equations in order to cancel terms. And so we're going to walk through an example where we do just that.

We're going to look at an example where first we have to multiply an equation by negative 1 before we add and cancel terms. And we'll walk through another example where first we have to multiply one of our equations by a scalar value before we can add them and cancel terms.

So before we get into that method, let's first quickly review the solution of a linear system. Well, so let's take 2x plus y equals 1 and x plus 2y equals 5. These are my two equations in my system.

Well, a solution would be a coordinate pair, a value of x and a value of y that satisfies both of these equations. Or more generally, if you have more than two equations, that would satisfy all of your equations. So you can see here if I substitute then negative 1 for x and 3 for y, I can write 2x plus y equals 1 as 2 times negative 1 plus 3 equals 1. And if we work this out, we get a true statement. It satisfies that equation.

If we substitute then negative 1 for x and 3 for y and our other equation, 2x plus 2y equals 5, we can work this out and get a true statement as well-- 5 equals 5. And graphically-- so I'm graphing these two lines. And the point negative 1, 3 is a point of intersection. It's an x value and a y value that satisfy all of the equations in our system.

Now there are several ways to solve for a linear system algebraically. We're going to be going over one of the strategies, which is called the addition method. So the addition method, what is the addition method?

We'll take our two equations here, 2x plus 3y equals 16 and 7x minus 3y equals 2. And I'm actually going to rewrite this so that I have one on top of the other. And what I notice here is a plus 3y and a minus 3y. So the addition method has us adding these two equations.

And when I add them together, my y term will cancel out because I have a positive 3y and a negative 3y. So what I'm going to get then is 2x plus 7x. I'll have 3y plus negative 3y. So that's kind of cancel out. I'll have nothing here.

And then I'll have 16 plus 2. So I have added these two equations together, and I have 9x equals 18. I only have one variable in this equation versus two variables.

So the addition method is also called the elimination method because I've eliminated this y term. So it's a strategy to solving a system of equations by adding equations in order to cancel variable terms. So with our addition method, we're adding equations, and we're eliminating a variable term. I have eliminated the y term.

So now I can easily solve for x by dividing both sides by 9. So x equals 2. How does this help me solve my system? I only know what x is. I don't know what y is.

Well, I can take any of these two equations-- it doesn't matter which one you choose-- substitute 2 for x and solve for y. So I'm just taking 2x plus 3y equals 16, plugging in 2 for x. So now I have 4 plus 3y equals 16.

Subtract 4 from both sides to get 3y equals 12. And finally, divide both sides by 3 to get y equals 4. So now I have x equals 2 and y equals 4. I can put this together as a coordinate pair. And the solution to this system is x equals 2 and y equals 4.

Now let's take a look at this example here where I notice that if I add these two equations, I'm not going to cancel out anything. I'd get. 6x and 2y. So I'm not canceling out anything. But I notice, hmm, if I could just subtract these two equations, that way, I'd be able to cancel out a term.

And we can actually call this then a subtraction method too. But an equivalent process then is multiplying one equation by negative 1. So when we do that, then, this is going to become negative 3x. This is going to become positive 2y. And this is going to be become positive 13.

So we're just changing the sign of everything. So this is really one way to rewrite a subtraction method to solving linear equations. But now when we add these two together, we cancel out that x term. So we have 6y equals 12.

So now we have a one variable equation we can easily solve for. We know that y equals 2. So once again, we can take any of our equations and plug in 2 for y and solve for x.

So now this would be 3x plus 8 equals negative 1. And subtracting 8 on both sides, 3x equals negative 9. So I know that x equals negative 3. So I know that x equals negative 3 and y equals 2. So the solution to this system is the points negative 3, 2.

Well, we're going to go through one more example where we want to then multiply one of our equations by a scalar value before adding. Because once again, I noticed that adding right off the bat isn't going to cancel anything. Subtracting isn't going to cancel anything either. So multiplying by negative 1 isn't going to help us.

But I do notice that if this were just 2x and a negative 2x instead of negative x, I would be able to add and then cancel out x term. So what I'm going to do then is multiply this entire equation by 2. So this will become negative 2x. This would become minus 4y. And this will become negative 6.

So multiplying by a scalar value, the goal is that you're changing your coefficients keeping everything equivalent. And then we would be able to add and eliminate. So now let's go ahead and add these two equations together.

We are eliminating then this x term. So we have 3y plus negative 4y. So this would be 3 minus 4 would give us a negative y equals 2. So if negative y equals 2, y equals negative 2.

Once again, we could take either of our two original equations in our system and then just substitute negative 2 in for y and solve for x. So now I have then 2x minus 6 equals 8 because 3 times negative 2 is a negative 6. I'll add 6 to both sides to get 2x equals 14.

And finally, divide both sides by 2 to reveal that x is 7. So now I have an x value and I have a y value. So solution to this system is 7 negative 2.

So let's review solving a system of linear equations using this addition method. So the addition method, we are adding equations in order to eliminate a variable term. So here's an example of how this works. We have a plus 2y and a negative 2y. So if we add these equations together, we can eliminate one of those terms, solve for x, and then substitute to solve for y.

Sometimes, you might have to multiply it by negative 1. This would also be thought as of a subtraction method too, noticing when straight up addition won't help. So first, you want to rewrite this so that you do have a negative term that you can cancel out through addition.

And sometimes, you might have to multiply it by a scalar value in order to change some coefficients to work in your favor. So here's an example of that. Multiplying this bottom equation by 2 so that when you add, you can eliminate one of those variable terms.

So that's it for solving a system of linear equations using this addition method. Thanks for watching.