given a system of two equations how can you tell if it's consistent or inconsistent if it's dependent or independent let's go over the important things they need to know let's just keep it simple so if you have a system of equation that has only one solution then it's going to be consistent and it's going to be independent if there's only one solution now sometimes you might have many solutions as opposed to one solution so if you have many solutions will it be consistent or inconsistent it turns out that it's still going to be consistent do you think it's going to be dependent or independent if there's many solutions it's dependent as opposed to independent now sometimes there's going to be no solution if there's no solution it's going to be inconsistent in addition it's going to be uh independent if there's no solution so make sure you know this information and that will go through a few examples using this uh information so how can we distinguish between one solution no solution and many solutions let's say if you solve a system of equations and you get one value for x and one value for one it's going to be one solution now what about no solution when you're solving it if you get to a point let's say like two equals five that is not a true statement so in a situation like this it's a no solution now what about many solutions how does that look like whenever you solve a system of two equations if you get something that looks like this zero equals zero five equals five or x equals x if the two sides are exactly the same then it's many solutions but if you get x equals a number rather than itself it's one solution so that's how you can distinguish between these three categories and once you know it's one solution you know it's consistent and independent if it's many solutions you know it's consistent and dependent if it's no solution it's inconsistent and now let's start with this example three x plus y is equal to seventeen and also four x minus y is equal to eighteen determine if there's one solution no solution or many solutions if it's consistent or inconsistent dependent or independent well let's use the elimination method to get the answer if we add the two equations y and negative y will cancel three x plus four x is seven x seventeen plus eighteen is thirty five now let's divide both sides by seven thirty five divided by 7 is 5. now that we have the x value let's plug it into the first equation to get the y value so 3 times 5 plus y is equal to 17. 3 times 5 is 15. and seventeen minus fifteen is two so y is equal to two so there's only one solution it's five comma two and whenever there's one solution is it going to be consistent or inconsistent one solution will always be associated with consistent and anytime you have one solution it's going to be independent it's always going to work out that way here's the next example 2x plus 4y is equal to 8 and also x plus two y is equal to four determine if this system of equations if it's consistent or inconsistent dependent or independent if it contains one solution no solution or many solutions well let's use elimination again let's multiply the second equation by negative two so first let's rewrite the first equation which is 2x plus 4y is equal to 8. now for the second equation x times negative 2 that's going to be negative 2x 2y times negative 2 is negative 4y 4 times negative 2 is negative 8. if we add the two equations negative 2x plus 2x is 0. 4y plus negative 4y is 0. 8 minus 8 is 0. so zero equals zero this is the case where we have many solutions now if there are many solutions then it's going to be consistent but dependent and so that's it for this problem try this one three x plus two y is equal to five and six x plus four y is equal to eight is there going to be one solution no solution or many solutions well let's multiply the first equation by negative 2. 3x times negative 2 is negative 6x 2y times negative 2 is negative 4y and 5 times negative 2 is negative 10. and let's rewrite the second equation so now if we add it negative 6x plus 6x it cancels and negative 4y plus 4y cancels so that's simply 0. negative 10 plus 8 is negative 2. now 0 does not equal negative 2. so this is the case where we have no solution and when there's no solution it is inconsistent it's inconsistent but it's independent as well and so that's it for this example