welcome to systems of linear equations substitution and elimination so first some definitions a system of equations is said to be consistent if it has at least one solution I always think of it using the idea that the solution set consists of something so it's either one or more a system of equations is said to be inconsistent if it has no solutions a system of equations is said to be independent if it has one solution exactly and the system of equations is said to be dependent if it has an infinite number of solutions so independent and dependent our terminology that are only applied to consistent systems of equations if a system is inconsistent then that's it it's not going to be classified any further as independent or dependent so a system that has one solution it's called consistent and independent a system that has no solutions is called inconsistent and a system that has an infinite number of solutions is called consistent and dependent so we're going to solve each of the systems of equations here and after we solve we're going to classify them based on their solution sets okay so this is a review from what you've studied before in algebra we have our two methods of either substitution or elimination and I'll do one at each for you to see so solve each system example a we have X plus 3y equals 5 and 2x minus 3y equals negative 8 well this one's just screaming for us to do the addition or elimination method right just adding the two equations together because the 3y and negative 3 or I will cancel out and now I have 3x is equal to negative 3 which means X is equal to negative 1 now I'm not done until I go find the corresponding Y value for the solution so you can substitute it into either equation 1 or 2 let's substitute it into one so I'll have negative 1 plus 3y is equal to 5 so 3y equals 6 which means y equals 2 now if you leave your answer like this you're not going to get full credit you need to write your answer in a solution set as an ordered pair so here's my solution set I have to write it as an ordered pair in parentheses and then now we're going to classify the system so I have exactly one solution that means it's consistent the solutions that consists of something and it's independent since I have exactly one solution all right now let's look at another example this time I'm going to solve using substitution I could still very easily solve using elimination the addition method but I just want to show you a different option so if we're using the substitution method you're gonna isolate one of the variables from one equation easiest thing is to isolate X here so x equals 5y plus 2 and now I'm gonna substitute that in for X into the other equation which means 3 times 5y plus 2 minus 15y equals 6 all right so now I have one equation with one variable so I can distribute the three I have 15 y plus 6 minus 15 y is 6 the 15 Y's cancel out and maybe you're getting a little nervous but don't be we're left with 6 equals 6 so anytime this happens you just have to ask yourself is this a true statement absolutely 6 is equal to 6 so that means we have an infinite number of solutions well how do you write your solution set no it's not all real numbers think about what kind of answers were giving here in the previous problem we gave an ordered pair so our answers need to be two-dimensional answers right and so all real numbers that we have a dimensional problem that refers to a one dimensional solution nor is my solution all of our to right r2 is the real number plain Cartesian XY plane two dimensions no not every single point that's here is a solution only the points that satisfy the equation X minus 5 y equals 2 since we have an infinite number of solutions means this equation is actually the same one if you notice carefully here the second equation is the first one multiplied by 3 so we have infinitely many solutions because any point that lies on that line is going to be part of my solution set so it's not any point in the XY plane but it's any point that satisfies the equation X minus 5y equals 2 well how do you write that out with great caution so you say your solution set is any ordered pair X Y such that it satisfies the equation X minus 5y equals 2 and I picked that equation just because it was on top you could just as easily have written the second equation and that would be fine so this is required for a full credit correct answer the solution set when you write X comma Y then you're letting your reader or your grater know that we're talking about a 2-dimensional answer such that and then the condition follows so this solution set is consistent and it's dependent since we have infinitely many solutions all right good now let's look at an application an airplane that flies from Los Angeles to New York with a stopover in Chicago charges a $45 fare to Chicago and a $60 fare from LA to New York a total of 185 passengers boarded the plane to LA and fares totaled ten thousand five hundred dollars I mean what a great deal for a flight I'm on it how many passengers got off the plane in Chicago okay so let's let X equal the number of passengers going to Chicago and then Y can be the number of passengers going to New York so if you have two variables you need two equations in order to solve so right away I see okay they told me I have a total of 185 passengers that x plus y equals 85 and then the next equation can describe the value of all of that so this first one talks about the quantity and then the next one will describe the value so each passenger going to Chicago each X pays $45 and each passenger going to New York pay $60 and that total is ten thousand five hundred dollars alright so I've got my two equations two variables now it's up to us how we want to solve substitution or elimination let's do substitution here so X is equal to 185 minus y and then now I'm going to substitute that for X into the other equation so I have 45 times 185 minus y plus 60 y equals 10,500 then I'm going to distribute so eight thousand three hundred and twenty-five minus forty five y plus 60 y is ten thousand five hundred combining like terms via 15 Y let's subtract over 80 325 that's 20 175 so y equals 145 okay but what did they want how many passengers got off the plane in Chicago so why is the passengers going to New York I still need X so if Y is 145 X is equal to 185 minus 145 which means X is 40 so 40 passengers got off the plane in Chicago okay looks good now let's kick it up a notch and look at how to solve a system of three linear equations so now we're gonna have three variables typically XY and Z or a B and C and in order to solve the system you need three equations so you want to write all equations in standard form variables on the Left constants on the right number the equations for later reference one two three clear any decimals or fractions it's optional but very helpful to do so now step three you want to select a variable to eliminate you use two of the equations to eliminate that variable and you call the result a now you're gonna use a different pair of equations and eliminate the same variable call the result be I'll talk to you I'll talk you through these two steps as we work through the following example so you can see and I really recommend that you write out like your game plan before you start doing those steps then you solve the system of equations a and B and then plug these values into one of the original equations to find the value of the third variable and then write your solution as an ordered triple I think so let's get started first example everything looks like it's in standard form so I'm just gonna start numbering the equations 1 2 & 3 and then here is where I know no fractions no decimals to clear out it's time to come up with our game plan for steps 3 & 4 you're gonna pick a variable that you want to eliminate okay so do you want to eliminate X or Y or Z it doesn't matter you just have to be consistent I would recommend either Y or Z because already if I just add some of the equations together I can see that they'll cancel out and the coefficients are 1 on both those variables in one of the equations so it's easier to work with so let's eliminate Z okay and then you have to pick two different pairs of equations that you're going to eliminate Z with and I want you to write out your game plan right now so we're gonna eliminate Z with how about 1 & 2 that looks good and then I'll eliminate Z again with maybe 2 & 3 you have to pick two different pairs of equations to do this with okay so let's start off by eliminating Z with equations 1 & 2 I don't need to do anything to them because I can just add the equations together and Z will cancel out right do you notice spot okay great so we're gonna add these equations together and now I get 6x bonus the Y canceled out I mean I wasn't hoping for that but that's fabulous so 6x is equal to 3 which means X is equal to 1/2 usually you won't already get a solution but we lucked out with this one so I'm gonna call this result a after you eliminate the variable with two of the equations that result is called a now I'm gonna move on to the next part of my plan and work with equations two and three to eliminate Z well notice here Z won't cancel out just by adding them together so what do I need to do I need to multiply equation two by two so I'm going to take equation two multiply it by two and now it's 4x plus 2y plus 2z equals negative 2 if you don't cross your Z's now it's a good time to start because let me tell you they're gonna get confused with your twos and then equation 3 is 6x minus 3y minus 2z equals 3 that needs to be a 3 okay so now let's add these equations together I'm gonna call the result equation B so I'm gonna have 10 X minus y disease cancel out as they should otherwise you messed up equals 1 now the equations a and B should have the same two variables so equation B has only X's and Y's right because we canceled out Z and same for equation a there's only an X if it had a Y in it to that would be no problem so now I solve this new system a and B so a house x equals 1/2 B has 10 X minus y equals 1 well I can already find Y because I'm just going to substitute in 1/2 for X so 10 times 1/2 minus y equals 1 so 5 minus y is 1 which means 4 is equal to Y so I have y I have X now I go back to any of the equations from the very original system and I find the last missing variable which is V okay well which equation do you want to work with it's up to you let's work with the equation 2 because it's just got a plus Z by itself so plugging everything into equation 2 so 2 is 2x plus y plus Z is negative 1 so I have 2 times X as 1/2 plus y is 4 plus Z is negative 1 so this is gonna be 1 plus 4 that's 5 plus Z is negative 1 so Z is negative 6 all right well how do I write my solution not like this you write it as an ordered triple so you have X first then Y then Z this is my solution set and then we're gonna classify it so I have a solution which means the system is consistent and there's exactly one solution so it's independent all right how are we doing if you're organized like this it'll work out if you just start using elimination and going wild with any set of equations that you want it spirals out of control and you end up with 10 new equations on your paper and you're no closer to getting to the solution so if you follow this method it will not serve you wrong ok now it frequently happens that one of the variables is missing from one or more of the equations and this tends to confuse students but it's actually helpful and it will save you some work so remember our approach is to transform the system into a system of two equations and two unknowns these are written in standard form I'm gonna call them equations one two and three like before and notice in the first equation B is missing and in the second equation a is missing so basically one the variables is already eliminated for us we just have to choose to eliminate that same variable again using the other two equations so do you want to eliminate B or a I'm gonna eliminate B if you chose a though that's fine okay so if I eliminate B that means equation 1 becomes my equation a because I've already eliminated be in that equation so I have a minus 3 C equals 6 well what two equations am I gonna use to eliminate B again so one was one of the choices and then I have to use two and three there's no other way to do it so let's see here if I take 2 and 3 and add them together B will not cancel out unless I take the equation 2 and I multiply it by 3 so if I multiply equation 2 by 3 it becomes 3 B plus 6 C equals 6 and then equation 3 is 7a minus 3b minus 5 C equals 14 all right so the 3b cancels out I'm gonna call this result equation B and here I have 7 a plus C equals 20 now notice I have a new system just focus on equation a and equation B and pretend this is one of the problems too from the very beginning of the section so we have a minus 3 C equals 6 and B which is 7 a plus C equals 20 all right easiest thing I think is to multiply equation B by 3 so now it's gonna become 21 a plus 3 C equals 60 equation a I need to change a minus 3 C is fix this together and now I have 22 a equals 66 whoo this is gonna work out nicely so a equals 3 hooray now what do I do next plug this back in to the very beginning only with caution don't plug it into equation 3 and it's of no use in equation 2 you got to plug it in to the first equation so that you can get C right okay so let's go back here into equation B now I have 7 times 3 plus C equals 20 so that's 21 plus C is 20 so C is negative 1 and then B plus 2 C equals 2 from Equation 2 so B plus 2 times negative 1 is 2 so B equals 4 do you leave your answer like this only if you want points off on your test so our final solution is the ordered triple 3 4 negative 1 and then let's classify it it's consistent and it's independent all right looks good now let's look at another example here again one of the variables is missing from one of our equations 1 2 3 so since Z is already missing in the third equation let's just choose to eliminate Z since half the work is done for me so we're going to eliminate Z I'm going to use equation 3 and then I'm going to use 1 and 2 to eliminate Z again so equation 3 becomes equation a which is 2x minus y equals negative 1 and then can I eliminate Z using 1 & 2 well if I take equation 1 as is and then if I take equation 2 and I multiply by negative 1 now it becomes X minus 2y minus C is negative 2 and now Z will cancel out and I'm left with 2x minus y equals negative 1 and this is equation B now look at what's going on here here's equation a here's the equations B and what is that tell us these are the same equation so I have infinitely many solutions well how do I write up my solution set it's a little more tricky you have to pick a variable to be your free variable and then you define all the other coordinates in terms of that variable what do I mean well you start from the last point here 2x minus y equals 1 pick a variable to solve for easiest variable to solve for is why don't you agree get it by itself so that means Y would equal 2x plus 1 okay that's great so why is 2x plus 1 that means I could say okay X is gonna be X Y it's gonna be 2x plus 1 now I need to figure out what Z is gonna be in terms of X how can I do that well go back here maybe using equation 1 I know X plus y plus Z equals 1 so using equation 1 I have X which is X plus y which is 2 X plus 1 plus Z equals 1 which means 3 X plus 1 plus Z is 1 so Z is equal to negative 3 X so now I have my ordered triple notice there's gonna be infinitely many solutions because X is my free variable X can be any real number and then based on what I choose for X I'm gonna get the corresponding ordered triple so here's what happens you write your solution set as X 2x plus 1 negative 3 X and X is any real number okay now you didn't necessarily have to solve for the other two coordinates in terms of X you could also do Y or Z but your answer may look just a little different however it will still generate the same solution set you'll still get the same infinite number of solutions and they'll be related in the same way so for example if I randomly chose X to be 1 then 1 comma 3 comma negative 1 is a solution to this system of equations and if I had a different free variable I'd still get the same ordered triple 1 3 negative 1 okay let's classify this bad boy so it is consistent we've got a solution we've got plenty of them and it's dependent because we have infinitely many dependent ok good last one can you guess what it's gonna be which kind have we not done yet so let's number our equations one two three pick a variable to eliminate how about we eliminate Y it's already missing in the first equation right so using equation 1 and then I'll eliminate Y again using the equations 2 & 3 so equation 1 becomes equation a which is X plus Z is 0 and then I need to eliminate Y using 2 & 3 so how about I multiply the equation 3 by negative 1 so the Y will cancel out so equation 2 I'll leave it alone X plus y plus 2z is 3 and then equation 3 I'm gonna multiply by negative 1 negative y minus Z equals negative 2 notice how I write exactly what I'm doing that way when you look back if you ever make a mistake you'll understand what steps you were taking in the moment you always know what's going on but when you look back sometimes you can't trace back what your reasoning was so the more notes you write yourself the more helpful it will be ok so Y cancels out like we wanted this is equation B and I have X plus Z equals 1 okay now let's look at equations a and B X plus Z equals 0 X plus Z equals 1 wait a minute how can X plus Z be 0 and then X plus Z be 1 at the same time that sounds very inconsistent can this system just make up its mind already my goodness that's because there's no solution and yes it's inconsistent what if you didn't notice say you were just in the zone and you were like I'm gonna multiply this second equation by a negative and add them together well then everything else would cancel out on the left right the X's would cancel disease would cancel you get 0 equals negative 1 does zero equal negative 1 not today that's false anytime you get a false statement same thing no solution inconsistent and these should be some of your favorite because then there's no work to do so when you get a true statement or everything cancels out then that's when you have a dependent system infinite number of solutions you've got to figure out how to write the solution set but if there's no solution and it's inconsistent then you're done okay so that concludes the lesson hope you enjoyed it and then coming soon our nonlinear system of equations