Systems of equations is a fundamental algebra topic. If 2x + 3y = 4 and 6x - 9y = 48, what do x and y equal?
One method of solving this system is through combination (also called elimination, and also called addition).
2x + 3y = 4
6x - 9y = 8
multiply the top equation by -3
-3(2x + 3y) = -3(4)
6x - 10y = 50
which gives
-6x - 9y = -12
6x - 10y = 50
the point of doing this is to have two variables that have opposite coefficients (in this case the x's).
We can then add up the equations
-6x - 9y = -12
+ 6x - 10y = 50
-19y = 38
if we divide both sides by -19, we get
-19y = 38
-19 -19
y = -2
if y= -2, we can plug that back in to either of the original equations to find x
2x + 3y = 4
2x + 3(-2) = 4
2x + (-6) = 4
+6 +6
2x = 10
2x = 10
2 2
x = 5
so x = 5 and y = -2. The point (5, -2) solves both equations.
But, let's look at that step where we added up the equations again:
-6x - 9y = -12
+ 6x - 10y = 50
-19y = 38
I always, after a couple days of doing this method, ask my students "did anyone wonder why we are allowed to add up equations? We've never done that before. Was there a moment where anyone thought if that was even allowed?" Usually one or two students admit that they briefly had that question (and it's usually a girl, truth be told).
Why is that allowed? It's kind of weird! Well, first things first, we aren't actually adding up the equations, we are adding up expressions, the two halves of the equations (6x - 10 is an expression, 50 is an expression, and they were equal to each other).
The Addition Property of Equality states that if a = b, then a + c = b + c, which basically means you can add whatever you want to both sides of an equation.
Well, let's look at that new equation we made by multiplying by -3:
-6x - 9y = -12.
I ask my students, can I add 4 to both sides?
-6x - 9y = -12
+4 +4 of course I can (some students don't think I can, in fact, because there was no -4
anywhere in the equation. I try to clear this up.)
so, could I add 6x, if I wanted to?
-6x - 9y = -12
+6x +6x yes, again, I could do that.
I could also add 3 to one side and 2 + 1 to the other, right?
-6x - 9y = -12
+3 + (2 + 1) as long as I am adding the same thing to both sides, then I can do anything.
Well, look at that other equation 6x - 10y = 50. The equals sign means that the two sides are the same. 6x - 10y is the same as 50. So really, if I want to add 50 to both sides of the first equation:
-6x - 9y = -12
+50 +50
I could change that first "50" into 6x - 10y (because they're the same thing! 6x - 10y = 50)
-6x - 9y = -12
+(6x - 10y) + 50
and at this point, it is easier to just add up the equations and save some writing.
-6x - 9y = -12
+ 6x - 10y = 50
-19y = 38
and then solve from there.
Does anyone care about this? Again, I would have maybe one or two per class who were interested. But understanding little things like this can really pay off in the long run. I was never taught this, not remotely. I was in a van driving to a frisbee tournament talking about math with a couple of friends, and some made the distinction between adding equations and adding expressions. I took some time to put the rest together and a lot of other things in math started to fall into place. I want to give this opportunity to my students as well. So I try to teach my students why.
One method of solving this system is through combination (also called elimination, and also called addition).
2x + 3y = 4
6x - 9y = 8
multiply the top equation by -3
-3(2x + 3y) = -3(4)
6x - 10y = 50
which gives
-6x - 9y = -12
6x - 10y = 50
the point of doing this is to have two variables that have opposite coefficients (in this case the x's).
We can then add up the equations
-6x - 9y = -12
+ 6x - 10y = 50
-19y = 38
if we divide both sides by -19, we get
-19y = 38
-19 -19
y = -2
if y= -2, we can plug that back in to either of the original equations to find x
2x + 3y = 4
2x + 3(-2) = 4
2x + (-6) = 4
+6 +6
2x = 10
2x = 10
2 2
x = 5
so x = 5 and y = -2. The point (5, -2) solves both equations.
But, let's look at that step where we added up the equations again:
-6x - 9y = -12
+ 6x - 10y = 50
-19y = 38
I always, after a couple days of doing this method, ask my students "did anyone wonder why we are allowed to add up equations? We've never done that before. Was there a moment where anyone thought if that was even allowed?" Usually one or two students admit that they briefly had that question (and it's usually a girl, truth be told).
Why is that allowed? It's kind of weird! Well, first things first, we aren't actually adding up the equations, we are adding up expressions, the two halves of the equations (6x - 10 is an expression, 50 is an expression, and they were equal to each other).
The Addition Property of Equality states that if a = b, then a + c = b + c, which basically means you can add whatever you want to both sides of an equation.
Well, let's look at that new equation we made by multiplying by -3:
-6x - 9y = -12.
I ask my students, can I add 4 to both sides?
-6x - 9y = -12
+4 +4 of course I can (some students don't think I can, in fact, because there was no -4
anywhere in the equation. I try to clear this up.)
so, could I add 6x, if I wanted to?
-6x - 9y = -12
+6x +6x yes, again, I could do that.
I could also add 3 to one side and 2 + 1 to the other, right?
-6x - 9y = -12
+3 + (2 + 1) as long as I am adding the same thing to both sides, then I can do anything.
Well, look at that other equation 6x - 10y = 50. The equals sign means that the two sides are the same. 6x - 10y is the same as 50. So really, if I want to add 50 to both sides of the first equation:
-6x - 9y = -12
+50 +50
I could change that first "50" into 6x - 10y (because they're the same thing! 6x - 10y = 50)
-6x - 9y = -12
+(6x - 10y) + 50
and at this point, it is easier to just add up the equations and save some writing.
-6x - 9y = -12
+ 6x - 10y = 50
-19y = 38
and then solve from there.
Does anyone care about this? Again, I would have maybe one or two per class who were interested. But understanding little things like this can really pay off in the long run. I was never taught this, not remotely. I was in a van driving to a frisbee tournament talking about math with a couple of friends, and some made the distinction between adding equations and adding expressions. I took some time to put the rest together and a lot of other things in math started to fall into place. I want to give this opportunity to my students as well. So I try to teach my students why.
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