[Music] welcome to miss miss math tutorials i'm miss smith in this video we're going to be talking about solving systems of linear and quadratic equations now we're both going to do this algebraically and graphically and then after we do it through graphing i'm going to show you how you can also do it on your calculator so let's dive right in so remember a system is two equations one will be linear so in this example this first one's our linear equation remember linear equations are to the power of one your highest exponent on a variable would be one and then a quadratic equation which in this case would be our second equation so its highest power on a variable is 2. what i want you to recognize that this is y equals 4x plus 1 and this is y equals x squared plus 4. so if y equals this and y also equals this then these two things equal each other so that's an equation that we can write so we can write 4x plus one equals x squared plus four if this equals this and this equals this then this equals this now at this point i want to get everything together on one side of the equation a lot of students are tempted to say okay well let's subtract x squared and move it over here technically you you can do that i don't recommend it i always recommend keeping the x square positive just to think just to keep things you know clean and simple all right so move the 4x by subtracting it over here whatever is the x squared side you want to go to that side now let's see so we've got at this point just a 1 equals x squared and no like terms here i can combine so it would just be negative 4 x plus 4. we always want to write our answer in standard form okay constant last of course i don't want this to be equal 1 i want it to be equal 0. so now i need to subtract 1 from each side so i've got zero equals x squared minus four x four minus one is positive three so now i've got this all combined together into one quadratic equation that i can solve okay so let's go ahead and label my a which would be one my b which would be negative four and my c which would be three now if you're not sure what i'm talking about when i say quadratic formula you definitely want to stop and go watch my video on using the quadratic formula before you watch this video remembering my quadratic formula remember that it is negative b negative negative 4 plus or minus square root of b so negative 4 squared you must have parentheses around that negative 4. if you do not your answer will be wrong okay so negative 4 square b squared minus 4 times a times c all over 2 times a remember when we use the quadratic formula we first simplify what we have underneath the radical so let's use our calculator to do that save some time negative 4 squared must have those parentheses minus four times one times three all right and we get four so that means we've got negative times a negative remember that's a positive 4 plus or minus the square root of 4 all over 2 times 1 is 2. okay so can we simplify square root of 4 because does that reduce is that a perfect square let's check square root of 4 and we get 2 we get a nice whole number so that reduces really well to 4 plus or minus 2 all over 2. so this is technically two separate statements written in one we have our four plus two over two that covers that positive and then we also have our four minus two over two okay so let's solve both of these and figure out what is x so four plus two is six six divided by two is three and then we have four minus two which is two two divided by two which is one okay so we know that our solutions here is x is the could be one and three okay so um here's the thing though we're not quite done because it's great to know that x is one and three but i wanna know the system i wanna know the actual intersecting points right i don't want to know just the x values i want to know the y values as well so we're going to want to take each of these x values back up to an original equation and figure out what will be my corresponding y values okay i can pick either equation i would pick the top one just because i don't want to deal with a square i'd rather just deal with a linear equation so one at a time i'm going to plug in each of these x values so first we'll do y equals four times x but first we're going to do one four times one plus one four times one is four plus one is five so i know that one of my solutions is one comma five i'm going to write my official answer just i don't know right down here so my final answer one intersecting point would be at one comma five and the other intersecting point i know the first the x would be three but let's figure out what the y would be so i would go back to the same equation y equals four times three plus one four times three is twelve and twelve plus 1 is 13. 3 comma 13. so those are my two intersecting points that is most definitely the long way around solving algebraically is a longer method to get your answer but it's something you do need to know how to do and the reason why you need to know some people might say well i'm always going to have a calculator with me well what if you get in a test so at some point in your math career and it's calculator inactive and they ask you to solve system of linear and quadratic equations you've got to know how to do it by hand as well i would never use this unless i have to it's good to know how to do it let's get to the shorter methods in terms of manually graphing this which we could do okay and i'm going to show you an even shorter way in a minute but let's go ahead and use our graphing calculator to help us first i need to graph this first line and then i will graph the second line now the first line i actually i don't even need the calculator i could graph this one by hand right it's just a linear equation so remember linear equations this is y equals mx plus b remember b is my starting point so it starts my y-intercept starts at positive one and then my slope tells me to move it's negative one so that means i move down one right one all right down one right one and i keep this pattern up down one right one so there's my linear equation now for my quadratic equation i definitely want to bring the calculator in to help me i would go to my y equals and type in negative x squared plus two x plus one okay now let's go to second graph to look at my table of values now i want to pick good values i definitely want to choose the vertex which remember that's a mirrored image right everything above it and below it is mirrored so it creates this kind of cool pattern effect so you want to definitely have the vertex in there which falls at one two let me go ahead and plot that one two and then we'll plot around it so then i've got 0 positive 1 0 positive 1 right there and negative 1 negative 2 negative 1 negative 2. and then we'll plot a little bit above the vertex two one three negative two all right and we'll connect those dots in our parabola form we want to physically see where i've got my linear equation i've got my quadratic equation where do they intersect well i can clearly see that they intersect right here at 0 1 and they intersect right here at three negative two okay so those are my two intersecting points now it's important to note that these could intersect twice like we see here it could only intersect once like what if i had a line and it just barely touched one point and then it just kept going okay so that would be an example of if they only had one solution or my linear line could be totally up here and not ever intersect that parabola and that would be no solution so you can have lots of different types of answers within those three parameters one solution no solution or two solutions those are your those are your three options okay now just a trick to quickly do this on the calculator i showed you the longest way the second longest way and now i'm going to show you the shortest way okay so assuming you do have a graphing calculator first thing you want to do is clear it so i've got a bunch of stuff in here from before second plus seven one two go and clear it i would wanna go to my y equals and type in both of these two equations now of course i need to make sure that they are both set equal to y first and they are so i can go ahead and plug them in the first one says negative x plus one the second one says negative x squared plus two x plus one and my order doesn't matter here right i could have done this one first and that one second it doesn't matter let's graph and let's look at these okay there's my linear and there's my quadratic and notice it looks just like what i what i graphed here so we'd be able to get kind of like a shortcut to that picture in order for me to find those intersecting points because i would want to know what those points are i just have to hit 2nd trace 5. and notice how one shows up right there a little i think of it as like a blinking spaceship it shows up right on the intersection all i have to do is hit enter three times one two three and it's going to tell me the intersection is at 0 and 1. and isn't that what i got 0 and 1. but now i want to know the other intersection because there are 2. i can clearly see 2 here so i just do it again i go second trace five this time i'm going to take my spaceship over to the other intersection right there and just get as close as you can one two three and see i see my other intersection falls at three negative two three negative two here's one for you guys to try on your own i've given you a linear equation y equals three x 1 and a quadratic equation y equals negative x squared plus 4x plus 1 whether you want to do it algebraically graphing it manually or graphing it on the calculator and i will tell you there are two for this one what are the two intersecting points for these two equations i will post the answer in the video description below this has been miss miss math tutorials you