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CCSS.Math:

scarlet and hensel's teacher gave them a system of linear equations to solve they each took a few steps that lead to the systems shown in the table below so is the teacher system this is what scarlet got after taking some steps this is what Hansel got which of them obtained a system that is equivalent to the teachers system and just to remind ourselves an equivalent system is a system that has or at least for our purposes is a system that has the same solution or the same solution set so if there's a certain X Y that satisfies this system in order for a scarlet system to be for to be equivalent it needs to have the same solution so let's look at this so scarlet let's see let's see if we can match these up so her her second equation here so this is interesting her second equation 14 X minus 7 y is equal to 2 over here the teacher has an equation 14 X minus 7 y is equal to 7 so this is interesting because the ratio between x and y is the same but then your constant term the constant term is going to be different and I would make the claim that this alone tells you that Scarlett's system is not equivalent to the teacher and you're saying well well how can how can I say that well these two equations if you were to write them into slope-intercept form you would see because the ratio between x and y the x and y terms is the same you're going to have the same slope but you're going to have different y-intercepts in fact we can actually solve for that so this equation right over here we can write it as if we let's see we if we subtract 14x from both sides you get negative 7y minus 14 whoops negative 7y is equal to is equal to negative 14x plus 7 and we could divide both sides by negative 7 you get Y is equal to 2x minus 1 so that's this all I did is algebraically manipulate this this is this line I could even try to graph it so let's do that so I'll draw a quick coordinate this is just going to be very rough quick coordinate axis right over there and then this line this line would look something like this so it's y-intercept is negative 1 and it has a slope of 2 so we draw something aligned with a slope of a line with a slope of two might look something something like that so that's this line right over here or or this one right over there and let's see this one over here is going to be if we do the same algebra we're going to have negative 7y is equal to negative 14x plus 2 or Y is equal to I'm just dividing everything by negative 7 2x minus 2/7 so this is going to look something like this its y-intercept is minus 2 7 so it's like right over there so this line is going to look something like draw my best my best attempt at drawing it it's going to look something it's going to look actually that's not quite right it's going to look something like I'll actually I'll just start right over here it's going to look something like something like this it's going to have the same slope and obviously it goes in it goes in this direction as well actually let me just draw that so it's going to have the same slope but a different different y-intercepts and that doesn't look right but you get the idea these two lines are parallel so these two lines are parallel ordinate that satisfies this one is not going to satisfy this one they have no points in common they are parallel that's definition of parallel since this and this have no points in common there's no way that some solution set that satisfies this would satisfy this because any XY that satisfies this can't satisfy this or vice versa and they're parallel there are no points these two things will never intersect so Scarlet does not have an equivalent system now what about what about Hansel what we see Hansel has the same thing going on here 5x minus y 5x minus y but then the constant term is different negative 6 positive 3 so this and this also represent parallel lines any XY pair that satisfies this there's no way that's going to satisfy this these two lines don't intersect they are parallel so Hansen's system is not and either