Eliminate or Substitute to solve this simultaneous equation?

well for guys in this video I want to cover how to solve a simultaneous equation using linear and a quadratic equation okay now one of the mistakes I see students make is they want to go ahead and apply elimination to a problem like this now you actually can do that what you'd want to do is actually rearrange them you're going to need to do is you're going to have to eliminate the Y variable because notice that even though I have X's here this is an x squared and this is a linear and this is an X so they're not like terms so therefore we can't eliminate the X you could if you wanted to eliminate the Y but in this example what I'm going to do is I'm actually going to use substitution and the reason why I like substitution is because I think it takes us to our third equation where we only have one variable much quicker if we're going to use the illumination method so what I'm going to do is I recognize here and again like the reason why substitution is usually going to be my preferred method here is again notice that I have a coefficient of 1 for my y right so whenever I see a coefficient of 1 for a variable immediately that comes to my brain of like I want to solve for that variable right and then apply the substitution process if I'm not dealing with the one or something times even a negative one then elimination might be something I would consider but in this problem let's just go ahead and solve for y and you can see solving for y was pretty simple right I just had to add the X to the both sides now again here's what's important here Y is equal to parentheses x plus 3. now why am I putting parentheses around there the reason why I'm putting parentheses around there is because it's one of the most common mistakes students will make is when they plug in X plus 3 in for this y they forget to put parentheses around it and if you forget to lose parentheses you're going to forget to use multiplication now first equation second equation now I'm going to do is I'm going to plug in the value of y which is X plus 3 into this second equation to now obtain a 30 question okay so what I want you to see is we had our original equation our second equation and now we create our third equation now again I manipulated equation number one right to solve for a variable but it was fairly simple then I plugged in equation number one into equation number three our equation number two to obtain equation number three now what's so important about equation number three well look how cool it is guys right again like now I only have X's so I know I can go ahead and solve this now the important thing is here as long as I applied my parentheses as long as I use these parentheses I know I'm going to have to apply my multiplication and now I can easily kind of simplify this out okay and now you recognize this is a quadratic and remember when we want to solve quadratics what we can do we need to set it equal to zero so go ahead and subtract the 9 to both sides and then it's going to be set equal to zero okay so now we have a quadratic equation set equals zero now I got to think to myself what two numbers multiply to give me negative 3 are going to add give me a positive two um that in the case I'm looking to add a positive 3 and a negative 1. okay so now what I did is I just used a zero product property set them both equal to zero and then went ahead and solved now a lot of times students will say like okay I'm done right but we got to think about like just don't automatically go to the answer I think understanding the graphical approach of what you're doing when solving simultaneous equations is very important remember guys this is a linear equation right I don't know what that graph looks like but let's just pretend you know it looks something like this and then this next example is a quadratic equation right if you were to solve for y here you would have some sort of quadratic now again I don't know what this quadratic looks like but let's just kind of write a hypothetical now is it possible that actually the quadratic and the linear equation could not intersect and we'd have no solution absolutely but in this case since we found our X and Y we know there's actually a solution that does exist it could also we could actually also have a point where that it intersects at a tangent line so it is possible to actually have an equation where there's only one solution but in this case we know there's not two solutions right we know there's two points of intersection but please do not just rely on like oh I found the X and that's like I'm done remember once you solve the X you've got to go and solve for y and here is why I prefer to use substitution in this problem rather than using elimination at the beginning because look what we did here guys for this equation number one right we already isolated it for y so half of my work now is already done for me so if I want to go ahead and take a look at my solution points if I want to find well what is the y coordinate right X is equal to 3 then what is y so all you're simply going to do is just take a y is equal to three plus three which equals A6 so now I have a coordinate point three comma six and what about when x equals negative one y equals negative one plus three right and then so therefore that's going to equal a oh wait I made a mistake here so this as actually factored wrong so that actually should be a positive 3 and this should be a negative one right because that's going to negative three okay so this is actually negative 3 and that's a positive one so therefore actually this is giving me a negative three plus three which is going to be a zero and in this case I'm going to add a 1 plus 3 is actually going to give me a 4. so now this coordinate point is going to be a one comma four make that a negative and there you go it's always good to be able to check your work and don't Factor so quickly because that's an easy mistake that you should not be making just like me but sometimes it happens so you just got to make sure you're aware of it and immediately sometimes always think and check your work in your brain as you're going through the problem now this problem was pretty cool recognizing a linear equation and a parabola but what about when you have two equations that are both going to be quadratic how would you solve a simultaneous equation like that that's coming up in the next video