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Solving a System of Linear Equations using the Addition Method

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Today we're going to talk about solving a system of linear equations using the addition method. Remember the solution to a system of equations is a specific coordinate pair, x and y, that satisfy all of the equations in the system. And there are several strategies for solving a system of linear equations.

Today we're going to talk about the addition method. The addition method is also called the elimination method, because we're going to be adding the equations in our system together to eliminate one of the variables. We'll go through some examples and show you how the addition method works.

For this first example, I'm going to solve this system of equations using the addition method. And in doing that, one of the variables, when I add the two equations together, is going to cancel out. We also call the addition method the elimination method, because when you add the equations together, one of the variables is going to eliminate.

And the way that you know which variable will eliminate, is the one that has the same coefficient in front of the variable. So both of these ys have a value of five, so we're going to eliminate the y-variable.

As I said, I'm going to add these equations together. So 3x plus 7x is going to give me 10x for my first term. 5y plus a negative 5y is going to give me 0y. And negative 9 plus 29 is going to give me 20.

So 0 times y is just going to give me 0. So this equation is really just 10x is equal to 20. So now I can solve this to get my x-variable. All I need to do is divide both sides by 10. Here the 10s will cancel, and I'm left with x is equal to two.

Now, to find my variable for y I can pick either one of my original equations, substitute my value for x into the equation, and find my value for y. I'm going to go ahead and just pick my second equation, 7x minus 5y equals 29, and again, substitute my value for 2 in for x.

So that's going to give me 7 times 2 minus 5y is equal to 29. So simplifying this, 7 times 2 will give me 14. Now I'm going to isolate my y-variable, so I'm going to subtract 14 from both sides. Here this will cancel, and I'm left with negative 5y is equal to 15. Then I'm going to divide both sides by negative 5, and I'm left with y is equal to negative 3.

So now that I have my value for y and my value for x, I can write my solution set as the point where the value for x is 2 and the value for y is negative 3.

For my second example, I have a system of equations that I want to solve using the elimination or addition method. And I see that my variable for x has the same coefficient of 6. However, if I were to add those together, positive 6 plus another positive 6 will give me 12, so that would give me 12x. And I want them to equal 0 so that the variable is canceled out.

So when we are adding our terms-- our equations together, we want the coefficient in front of the variable that we want to eliminate to have opposite signs, so one positive and one negative. So to achieve that, I'm going to multiply one of the equations by negative 1, so that the audience 6x term in one of the equations will be negative. I'm going to go ahead and pick the second equation and multiply all the terms in the equation by negative 1.

So this will become negative 6x plus 5y is equal to positive 14. I'm going to go ahead and cross out this middle equation. And so I'm going to solve my system by adding these two equations together.

So when I do that, 6x plus a negative 6x is going to give me 0x. 3y plus 5y will give me 8y. And 18 plus 14 is going to give me 32.

So now 0x is equal to 0, so the x-variable is cancelled out, and I just solve 8y is equal to 32. I'll divide by 8 on both sides. And I find that y is equal to 4.

Now I can pick either one of my original equations, or any of my original equations, substitute 4 in for my y-variable, and determine what x is. I'm going to go ahead and use my first equation. So I will have 6x plus 3 times 4 is equal to 18. Simplifying 3 times 4 is equal to 12.

Then I'm going to subtract 12 from both sides to cancel that out. And I'm left with 6x is equal to 6. Then I'll divide both sides by 6, and I find that x is equal to 1. So I can write my solution of x is equal to 1, and y is equal to 4 as a pair of points. And that would be 1, 4.

For my last example, I've got another system of equations. And I want to again solve using the addition or elimination method. But I see here that neither my x-variable nor my y-variable have the same coefficients. So similar to the way that we multiplied by negative 1 in our last example, we can multiply one of the equations by a constant value, so that either the x-variable or the y-variable has the same coefficient with opposite signs.

I could either multiply this second equation by a negative 5, in which case this term would become negative 5x and my x-variable would cancel. Or I could multiply my first equation by a positive 2, in which case this term would become 2y and then when added, my y-variables would cancel.

I'm going to go ahead and multiply my second equation by negative 5. So this equation, when I multiply all the terms by negative 5, will become negative 5x. Negative 5 times negative 2 will give me positive 10y is going to be equal to a positive 70.

Now I'm going to add these two equations together. And doing that I have 5x plus a negative 5x, which is going to give me 0x. y plus 10y will give me 11y, and 7 plus 70 is going to give me 77. 0x is just 0, so this becomes 11y equals 77. And dividing both sides by 11, I have that y is equal to 7.

Now that I have my y-variable, I'm going to pick one of my original equations and substitute the value of 7 in for y to find my x. I'm going to use my first equation. I have 5x plus 7 is equal to 7.

I'm going to subtract 7 from both sides, to cancel this out. And I have 5x is equal to 0. Then I'll divide both sides by 5, and I find that x is equal to 0. So I can write this solution of x equals 0 and y equals 7 as a coordinate pair, which would be 0, 7.

Let's go over our key points from today. Make sure you have them in your notes if you don't already, so you can refer to them later. A solution to a system of equations is a specific coordinate pair, x, y, that satisfies all equations in the system. The addition method, also called the elimination method, involves adding equations together to cancel out or eliminate one of the variables. And you may need to multiply one or both of the equations by a number so that when the equations are added, one of the variables will be eliminated.

- Addition method:
- Also called the elimination method, a strategy to solving a system of equations by adding equations in order to cancel variable terms.

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