Solving Systems of Equations using Elimination - Problem 2
This is a problem where I have two equations and two unknowns I'm asked to solve it using elimination. To do elimination I want to look at the coefficients or Xs and Ys and look for additive inverses.
But the way this is written right now it's kind of confusing for me because they are not lined up properly. I'm going to just rearrange the Xs and Ys in the second equation so that my -2x will come first.
Here's what I mean the first equation is going to stay exactly the same, the second equation I'm just going to rewrite it so that the 2x or the -2x term comes first.
That's going to make it easier for me to work with the Xs and Ys and their coefficients. So a lot of students make what I'm just about to do I'm going to make a mistake. I know so I'm going to show you where I see my Math classes all the time students say okay I'm going to eliminate that's where I add vertically. So they go ahead and they add vertically and they get to something like this they think they are on their way to an A+. But then they say wait a second I'm totally stuck then they say, "Wait teacher what do I do next?"
Here's is the thing you guys the whole point of elimination is to get one of your variables to be eliminated, nothing got eliminated here. Before I can add this vertically I need to do something to the coefficients so that I have additive inverses.
Look at the Xs I have -2 and -2 what I want to have is -2 and +2. So if I can do something to this equation so that instead of having a -2 right there I have a +2, I'll be in business.
In order to change that from a negative sign to a positive sign I'm going to have to multiply this entire second equation by -1. The top equation will stay the same. -2x plus 7y equals 11 I haven't changed anything yetm but in the second equation I have +2x take away 3y equals -19 and now I'm ready to eliminate because now when I add vertically I have 0 Xs. 4y equals -8, y is equal to -2 and I'm almost there. That's what my answer is going to look like I still need to find the x coordinate.
In order to find the x coordinate that goes in there I'm going to need to substitute my y value of -2 into either original equation. I'm going to use the first one but if I do everything correctly it shouldn't matter, I should get the same answer with the top or bottom equation. So come on pen help me out here, here we go.
2x plus 7 times -2 is going to equal 11. Let's go through and solve for x, add 14 to both sides. Oh! No I'm going to get a nasty fraction. Don't worry sometimes you get fractions in Math class it's okay. Divide both sides by -2 and I'll have -25/2 for my x coordinate.
A lot of times you do get nasty fractions in math class but luckily we have an easy way to check when it comes to systems of equations. The way you check is by going back to both original equations plugging in your xy coordinate pairs and making sure you get true equalities.
Here's what I mean -2 times -25/2 plus 7 times my y value should hopefully equal 11 let's check. Those negatives become a positive, those 2's cancel out so I have +25 take away 14 equals 11? Good it does, that means I checked in my first equation. Let's check the bottom original equation.
3 times my y value take away 2 times -25/2 hopefully it's going to give me my answer 19 coming from that equation. Let's make sure -6 plus 25, good does indeed equal 19, that means I did it right.
So you guys a lot of times you are going to get a lot of fractions like points are not always whole integers, sometimes you get fractional values like this. Just make sure you remember how to check your work by substituting in your xy coordinate pairs to both equations and make sure you have equalities.