Number of solutions to equations

determine the number of solutions for each of these equations and they give us three equations right over here and before I deal with these equations in particular let's just remind ourselves about when we might have one or infinite or no solutions you're going to have one solution if you can by solving the equation come up with something like X is equal to some number let's say X is equal to if I want to say in the abstract X is equal to a or if we actually were to solve it we get something like x equals 5 or 10 or negative pi whatever it might be but if you could actually solve for specific X then you have one solution so this is one solution just like that now if you go and you try to manipulate these equations in completely legitimate ways but you end up with something crazy like three equals five then you have no solutions no solutions and if you just think about it reasonably all of these equations are about finding and finding an X that satisfies this and if you were to just keep simplifying it and you were to get something like three equals five and you were to ask yourself the question is there any X that can somehow magically make three equal five no no X can magically make three equal five so there's no way that you can make this thing be actually true no matter which X you pick so if you get something very strange like this this means there's no solution on the other hand if you get something like five equals five and I'm just over using the number five it didn't have to be the number five it could be seven or ten or or 113 whatever and like see let me just not use five just to make sure that you don't think it's only four five if I just get something that something is equal to itself which is just going to be true no matter what X you pick any X you pick this would be true for well then you have an infinite infinite solutions so with that is a little bit of a primer let's try to tackle these three equations so over here let's see maybe we could subtract if we want to get rid of this two here on the left-hand side we could subtract two from both sides if we subtract two from both sides we are going to be left with on the left-hand side we're going to be left with negative 7x and on the right hand side you're going to be left with 2x this is going to cancel - 9 X 2x - a 9x if we simplify that that's negative 7x so you get negative 7x is equal to negative 7x and you probably see where this is going this is already true for any X that you pick negative 7 times that X is going to be equal to negative 7 times that X so we already are going into this scenario but you're like eh so I don't see 13 equals 13 well what if you did something like you divide both sides by negative 7 negative 7 at this point what I'm doing is kind of unnecessary you already understand that negative 7 times some number is always going to be negative 7 times that number but if we were to do this we would get X is equal to X and then we could subtract X from both sides and then you would get 0 equals zero which is true for all for any X that you pick 0 is always going to be equal to 0 so any of these statements are going to be true for any X you pick so this for this equation right over here we have an infinite number of infinite number of solutions let's think about this one right over here in the middle so once again well let's try two I'll do a little bit different I'll add this 2x and this and this negative 9x right over there so we will get negative 7x plus 3 is equal to is equal to negative 7x so 2x + 9 X is negative 7x plus 2 plus 2 well let's add X we've done a green color let's do that in that green color plus 2 that's this 2 now let's add 7x to both sides well if you add 7x to the left hand side you're just going to be left with the 3 there and if you add 7x to the right hand side this is going to go away and you're just going to be left with the 2 there so all I did is I added 7x I added 7x to both sides of that equation and now we got something nonsensical I don't care what X you pick how magical that X might be there's no way that that X is going to make 3 equal to 2 so in this scenario right over here we have no no solutions there's no X in the universe that can satisfy this equation now let's try this third scenario so once again let us maybe we'll subtract 3 from both sides just to see get rid of this constant term so we're going to get negative 7x on the left hand side on the right hand side we're going to have 2x minus 1 2 X minus 1 and now we can subtract 2x from both sides to subtract 2x from both sides you're going to get so subtracting 2x you're going to get negative 9 X is equal to negative 1 now you can divide both sides by negative 9 and you are left with X is equal to 1/9 so we're in this scenario right over here we very explicitly we're able to find an X x equals 1/9 that satisfies this equation so this right over here has exactly one one solution