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Systems of equations with substitution: y=-5x+8 & 10x+2y=-2

CCSS.Math: ,

Video transcript

use substitution to solve for x and y and then give us a system of equations here y is equal to negative 5x plus 8 and 10 X plus 2y is equal to negative 2 so they've set it up for us pretty well they already have Y explicitly solved for up here so they tell us this first constraint tells us that y must be equal to negative 5x plus 8 so when we go to the second constraint here every time we see a Y we say well the first constraint tells us that y must be equal to negative 5x plus 8 so everywhere we see Y we can substitute it with negative 5x plus 8 because that's what the first constraint tells us Y is equal to that I don't want to be repetitive and I really want you to internalize that's all it's saying why is that so every time we see Y in the second constraint we can substitute it with that so let's do it so the second equation over here is 10 X plus 2 and instead of writing a Y there and I've said it multiple times already we can write a negative 5x Plus 8 the first constraint tells us that's what Y is so negative 5 X plus 8 is equal to negative 2 now we have one equation with one unknown we can just solve for X we have 10 X plus so we can multiply it we can distribute this 2 onto both of these terms so we have 2 times negative 5x is negative 10x and then 2 times 8 is 16 so plus 16 plus 16 is equal to negative 2 now we have 10x minus 10x those guys cancel out 10x minus 10x is equal to 0 so these guys cancel out and we're just left with 16 equals negative 2 which is crazy we know that 16 does not equal negative 2 this is an inconsistent result and that's because these two lines actually don't intersect and we could see that by actually graphing these lines whenever you get something like some number equaling some other number that they're clearly not equal to that means it's an inconsistent result it's an inconsistent system and that these lines actually don't intersect so let me just graph these just to make it clear this this first equation is already in slope y-intercept form so it looks something like this that's our x axis this is our Y axis and it's negative 5 X plus 8 so 1 2 3 4 5 6 7 8 and then it has a very steep naked on word slope as every time you move forward 1 you have to go down 5 so it looks something like that that's this first equation right over there the second equation let me rewrite it in slope y-intercept form so it's 10 X plus 2y is equal to negative 2 let's subtract 10x from both sides you get 2 y is equal to negative 10 X minus 2 let's divide both sides by 2 you get Y is equal to negative 5 X negative 5x minus 1 so it's y-intercept is negative 1 it's right over there and it has the same slope as this first line so it looks like this it's parallel it's just shifted down a bit so it just looks like that so they're parallel lines they have the same slope different y-intercept we get an inconsistent result inconsistent inconsistent results they don't intersect and the tell-tale sign of that when you're doing it algebraically is you get something wacky like this this is why it's called inconsistent it's not consistent for 16 to be equal to negative 2 these don't intersect there's no solution to both of these constraints no x and y that satisfies both of them