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Topic D: Systems of linear equations and their solutions

Video transcript

solve using the elimination method and they tell us the sum of two numbers is 70 their difference is 24 what are the two numbers let's use this first sentence let's construct an equation from this first sentence let's construct a constraint the sum of two numbers let's call those numbers X and y so there's some X plus y is equal to 70 that's what this first sentence tells us the second sentence says their difference is 24 so that means that X minus y is equal to 24 we're just going to assume that X is the larger of the two numbers and Y is the smaller one so when you take the difference like this you get positive 24 so we have a system of two equations with two unknowns and they want us to solve it using the elimination method so let's do that so we can literally just if we were to just add these two equations we would on the left-hand side we would have a positive Y and we would have a negative Y over here and they would cancel out so if we were to just add these two equations we would be able to eliminate the Y's so let's do that so X plus y plus X minus y well the plus y and the minus y cancel out and you're just left with an X plus an X which is 2x and then that is going to be equal to 70 plus 24 70 plus 24 is 94 and I want to make it very clear and I mentioned it in previous videos that this process of adding the equations to each other this is nothing new we're really just adding the same thing to both sides we could view it as to both sides of this equation you could say we're adding 24 to both sides of this equation over here we're explicitly adding 24 to the 70 and over here you could say we could add 24 to X plus y but the second constraint tells us that X minus y is the same thing as 24 so we're adding the same thing to both sides here we're calling it 24 here we're calling it X minus y and we were able to eliminate the Y so we get 2x is equal to 94 now we can divide both sides of this equation by 2 and we are left with X is equal to what is that 40 47 and now we can substitute back in to either one of these equations to solve for y so let's try this first one over here so we have 47 plus y is equal to 70 can subtract 47 from both sides of this equation so we subtract 47 and we are left with y is equal to what is this 23 y is equal to 23 and you can verify that it works if you add the two numbers 47 plus 23 you definitely get 70 and then if you take 47 minus 23 you definitely get 24 so it definitely meets both constraints