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Topic D: Systems of linear equations and their solutions

Video transcript

so that we don't get stumped by the our bagel as in our life and especially when we don't have talking words to help us we should be able to identify when things get a little bit weird with our systems of equations when we have scenarios that have an infinite number of solutions or that have no solutions at all and just as a little bit of a review of what could happen these are the think about the three scenarios you have the first scenario which is kind of where we started off where you have two systems that just intersect in one place and then you have essentially one solution so if you were to graphically represent it you have one solution right over there one one solution and this means that the two constraints are consistent consistent and the two constraints are independent of each other they're not the exact same line consistent in the pendant consistent and independent then you have the other scenario you have the other scenario where they're consistent they intersect but they're essentially the same line they intersect everywhere so this is one of the constraints or one of the equations of the other one if you look at it if you graph it is actually the exact same one so here you have an infinite number infinite number of solutions it's consistent it's consistent you do have solutions here but there are dependent equations it's a dependent system dependent and then the last scenario and this is when you're dealing in two dimensions the last scenario R is where your two constraints just don't intersect with each other one might look like this and then the other might look like this they have the exact same slope but they have different intercepts so this there is no solution no solution they never intersect and we call this an inconsistent system in consistent and if you wanted to think about what would happen just think about what's going on here here you have different slopes here they have different slopes different slopes and if you think about it two different lines with different slopes are definitely going to intersect in exactly one place in exactly one place here they have the same slope and same y-intercept same same slope and same y-intercepts you have an infinite number of solutions here you have the same slope same slope but different y-intercepts and you get no solutions so the times when you're solving systems where things are going to get a little bit weird are where you have the same as when you have the same slope and if you think about it what defines the slope and I encourage you to test this out with different equations is when you have if you have your X's and Y's where you have your A's and B's or you have your variables on state on the same side of an equation where they have the same ratio with respect to each other so with that kind of keeping that in mind let's see if we can if we can think about what types of solutions we might find so let's take this down so they say determine how many solutions exist for the systems of acquit system of equations so you have 10 X minus 2y is equal to 4 and 10 X minus 2y is equal to 16 so just based on what we just talked about the X and the XS is and the Y's they're on the same side of the equation and the ratio is 10 to negative 2 10 to negative 2 same ratio so something strange is going to happen here but when we have the same kind of combination of X's and Y's in the first one we get 4 and on the second one we get 16 so that seems a little bit bizarre another way to think about we have the same number of X's the same number of Y's but we got a same a different number on the right-hand side so if you were to simplify this and we could even look at the hints to see what it says you'll see that you're going to end up with the same slope but different y-intercepts so we convert both the slope-intercept form right over here and you see one the blue one is y is equal to 5x minus 2 and the green one is y is equal to 5x minus 8 same slope same ratio between the X's and the Y's but but you have different values right over here you have different different y-intercepts so here you have no solutions no solutions that is that is this an Aereo right over here if you were to graph it so no solutions check our answer let's go to the next question so let's look at this one right over here so we have negative five times X and negative one times y we have four times X and 1 times y so it looks like the ratio if then we're looking at the X's and Y's always on the left hand side right over here it looks like the ratios of X's Y's are different you have essentially five X's for every one y you can say negative five x is for every negative 1y and here you have four X's for every one Y so this is fundamentally a different ratio so right off the bat you can say what this these are going to intersect in exactly one place if you were to put this into slope-intercept form you will see that they have different slopes so you could say this has one solution and you can check your answer and you can look at the solution just to verify so and I encourage you to do this so you see the blue one if you put in the slope intercept form negative 5 X plus 10 and you take the green one into slope intercept form negative 4 X minus 8 so different slopes they're definitely going to intersect in exactly one place you're going to have one solution let's try another one so here we have 2 X plus y is equal to negative 3 and this is pretty clear you have two X plus y is equal to negative 3 these are the exact same equations so that it's consistent information there's definitely solutions but there's an infinite number of solutions right over here these are this is a dependent system so there are infinite number of solutions here we can check our answer let's do one more because that one's a little bit too easy okay so this is interesting right over here we have it in different forms 2x plus y is equal to negative 4 y is equal to negative 2x minus 4 so let's take this first blue equation let's take this blue equation and put it into slope-intercept form if we did that you would get if you just subtract 2x from both sides you get Y is equal to negative 2x minus 4 which is the exact same thing as this equation right over here so once again they're the exact same equation you have an infinite number of solutions check our answer you can look at the solution right here you convert the blue into slope-intercept you get the exact same thing as what you saw in the green one