[Music] in this video i want to take a look at system of equations on the sat exam in particular i want to talk mostly about what it means when you have no solution and what it means when you have an infinite number of solutions so just in general um why don't we just talk about what does it mean for a system of equations to have a solution well let's think about this from a graphical standpoint here okay if for instance i were to just graph two lines along the y and x axis okay one and two okay it would look something like this a solution would be where those two lines intersect okay it's going to be where they have the exact same x and y values so if a system of equations has a solution means it's where they intersect well then what do you think it means when we say a system has no solution well if a system has no solution that means the lines are never going to intersect so never intersect and what we want to think about this is think about then most of the time you're going to be dealing with linear systems on this test meaning straight lines so what kind of straight lines never intersect well we call those lines parallel and parallel lines what we know about them is that they have equal slopes or the same slopes okay well so if no solution means that our lines never intersect then we gotta say okay well what do you think infinite solutions means infinite solutions well infinite solutions means that they always intersect and if they're always intersecting and again we're talking about a linear system okay then they have to be the same line which means they are the same line and if they're the same line well then they must have the same slopes okay now you'll notice in both these situations whether it's no solution or whether it's infinite solution for both of these we're saying that the slopes of our lines must be the same as long as it's a linear system here so let's take a look at some of these examples and let's kind of apply this idea here so take a look at question number one it says if the equation above has an infinite number of solutions what is the value of m now some people get confused with just kind of the way this is uh written here so you might want to think of this as like two equations like this is like y1 and the right side is going to be y2 here okay so people are used to seeing it like that um but first you should also make sure to distribute this out here so that first one would become two m x plus um eight m minus three and then the second equation okay down here y2 that would be 6x plus 21. now if those are our two equations for our linear system here well if they have an infinite number of solutions they must have the same slope so the slope of the first one here okay we said was 6 okay or the second one i should say and the slope of the first one is 2m it's everything that's right in front of the x value there so here in order for those to be the same we know that m must be 3 because 2 times 3 would give me 6. and that's it you can make sure they have the same slope okay let's take a look at question 2 here so question 2 it says if the system of equations above has an infinite number of solutions what's the value of a over b now i'm going to show you two ways to do this because there is a shortcut method you can use that will save you a ton of time on this test but i want to show you why this works here okay say i wanted again infinite number of solutions means they have the same slope so i'm going to put them in slope-intercept form here so i'm going to rearrange that top equation by subtracting the 2x from both sides so we get 5y is equal to negative 2x plus and then simply just dividing everything by 5. so we get that this y is negative two-fifths x plus 12. okay and then i do the same thing for that second equation here okay so if i have ax plus b y is equal to 20 i'm going to start by subtracting off ax okay so we get b y is equal to negative ax plus 20 and then i'd simply just divide by b okay so there's our equation for the second one y equals negative a over b x plus 20 over b so since it has infinite solutions their slopes must be the same so the slope of the first one is negative two-fifths slope of the second one here is negative a over b so those must be the same negative two-fifths has to equal negative a over b and then if i just divide both sides by negative one that's going to make both these positive so a over b is 2 over 5. and there's your answer so now the shortcut for this is this if you ever come across a question like this and says infinite solution or no solution for a problem like this and everything's lined up like the x terms lined up the y terms lined up and the numbers are on the other side of the equal sign and the equal signs are lined up if everything's lined up you can always do this trick since their slopes are the same that means the ratio of the coefficients between the x terms and the y terms also has to be the same so i could literally just read off here and say 2 is to 5 2 is to 5 as a is to b as a is to b and you have your answer right off the bat here all right um let's take a look at number three this is one that a lot of people get confused with it says if the expression ax squared plus 3ax minus 2x squared is equivalent to bx then what is the value of b okay so let's write that out ax squared plus 3ax minus 2x squared is equivalent to bx okay so what i want to do on this is i like to balance the equation and what i mean by that is if i have x squared terms on the left side i want to make sure i have x squared terms on the right side or if i'm an x term and left i want to make sure i have an x on the right and so forth and so forth so what i'm going to do here is i'm going to actually add this 2x squared over to the other side here so then i get ax squared plus 3ax equals 2x squared plus bx okay well now looking at this if i have 2x squareds on the right then i better have 2x squared also on the left and the only way i can have 2x squareds on the left is if a right here was equal to 2. so i know that a must be equal to 2 so we get 2x squared plus and then 3 times 2 would just be 6 x equals 2x squared plus bx okay well now we have 2x squared is equal to 2x squared so then we just have to finish this out then then in order to get the b value bx must be equal to 6x so b would have to be 6. and there's our answer okay so just make sure the left and right hand side are the exact same they're going to be equal to one another okay number four we're going to use that shortcut that we saw for problem 2 here so it says if the system of equations above has no solution then what is the value of k all right so for this problem here again since everything's lined up our x terms lined up our y terms lined up and the numbers are lined up with along with the equal sign no solution again means that they have the same slope which means the ratio of their coefficients are equal so i can literally just say 3 is related to negative 4 as 4 is related to negative k so i've set up this ratio here and now i can just cross multiply and solve so 3 times negative k is negative 3k 4 times a negative 4 is equal to negative 16 and then just divide by negative 3 and so we get that k is equal to 16 thirds choice c all right uh let's go down here to number five so number five it says in the system of equations above a and c a b and c are constants if the equation has infinitely many solutions then which of the fine must be equal to c okay so here we go again we have infinitely many solutions infinitely many means they have to be the exact same line which again means that their slopes have to be the same so if i distribute this out we get that this is 3x plus 3a and then if i go to the right hand side there we get bx plus c well infinitely many they have to be the same and they have to have the same slope then b here must be equal to 3 because if i have 3x and i have bx then b has to be 3. okay well if that's true then and i write this down okay 3x is equal to 3x already so then in order for them to be the exact same c would have to be 3a or choice c there that's it okay let's take a look at 6 then on the right hand page here so this one says in the equation above b is a constant if the equation has no solution what's the value of b again no solution means that our lines just have to have the same slope so if i look at the left hand side here the slope of that is just 5. well if i have a slope of 5 on the left then i better have a slope of 5 on the right now the slope on the right is denoted by this letter b here okay it's everything in front of the x value so that means that b must also be 5 in order for those to be the same and so that's choice d okay number seven this one gets a little tricky this doesn't have so much to do with infinite or no solution but just kind of a tricky system question i want to go through here and this would be on the non-calculator part of the test it says given the system of equations above what is the value of 35x plus 14y now you might get a problem where it says something weird like this 35x 14y that just sounds really really weird here so what i suggest doing is factoring that out factoring something out here well 35x and 14y i can factor out a 7 and then i can get 5x plus 2y okay when you factor out that gcf of 7 here well if that's the case you want to look to try to force this to happen can i get 5x plus 2y by doing something to these two equations up here well say wait 5x i can get 5x simply by adding together 3x and 2x so that's what i would do here i just add these together so 3x plus 2x is 5x negative 4y plus 6y would have just be a positive 2y and then 3 plus 5 is just 8. oh 5x plus 2y well here is 5x plus 2y 5x plus 2y is 8. so we can just substitute this in for 8. so we're left with 8 times 7 which is 56 and that's why this is in the non-calculated part of the sat because you can force something like that to happen okay same thing you might see it ask like what's 10x plus 10y or 5x plus 5y okay usually you can force that to happen by either adding or subtracting your two equations together so just be aware of that number eight just want to make sure that you're okay and know how to solve a system normally okay it says given the system of equations above what's the value of x minus y uh very simple for this one um you're just going to solve this i break and there's a bunch of ways but i would recommend using the elimination method so i see that uh we have a positive y here and a negative 2y on top so i'm going to eliminate the y values simply just by multiplying this bottom equation by 2. so the top equation will stay 4x minus 2y is 8 but if i distribute this 2 to the bottom here that's going to become 6x plus 2y equals 2. and now i'm just going to add these together so 4x plus 6x is 10x negative 2y plus 2y gets eliminated okay that's why we call this elimination method bring down the equal sign 8 plus 2 is going to be 10 and so we simply just divide by 10 and we get that x is 1. so there's my x value now i can just take that and put that back into either equation to get the y value so i'm going to put into that second equation and we get 3 times one plus y is equal to one three plus y is equal to one subtract off the three from both sides and we get that y is negative two so what's the value of x minus y well that's one minus negative 2 and minus a negative becomes a positive 1 plus 2 is 3. and there you go so that's system of equations which is one of the most heavily tested topics on the test if you do have more questions on this or there's anything else you would like to see feel free to leave a comment below