Someone please explain in the most simple way possible please. (edit: i am using elimination)
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In: 1
When you write an equation like x + 7 = 9, you’re saying “there’s some unknown number, which I’m going to represent with an x. I don’t know what it is yet, but I know if I add 7 to it, I get 9”. And solving the equation is trying to find values of x that would make that statement true. x=2 is a solution because 2 + 7 = 9 is a true equation.
Simultaneous equations are the same idea, except that you have *multiple* pieces of information about your variables and (typically) more than one variable. So if, for example, you have:
* -2x + 3y = 7
* 3y = 9
then what I’m saying is:
* There are two numbers I don’t know. I’m writing one of them as x and the other as y.
* I know that if I take minus two times x, and add 3 times y, I get 7.
* I know that if I take three times y, I get 9.
* I want to know what x and y are.
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There is a general fact in math that is very helpful in solving equations:
* If you take a true equation, and do the same thing to both sides, you still have a true equation. In particular, any value of the variables that solved the first equation will solve the new one, too.
And in particular:
* If you take a true equation, and do the same **reversible** thing to both sides, not only do you still have a true equation, but the solutions to the new equation are the same as the solutions to the old equation.
Reversible operations include things like adding, subtracting, or multiplying or dividing by anything but zero. And in this case, those are the operations we’re going to use.
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So. We have our equations, and our goal is to transform them into equations we know how to solve. There are a few ways to go about this, but you specified elimination, so we’ll use that.
Our goal is to take the equations involving *two* variables and turn them into an equation involving *one* variable, because we know how to solve one-variable equations.
Let’s consider the first equation, -2x + 3y = 7. Wouldn’t it be nice if we could just get rid of that 3y somehow, so we just had something with *x*? We can subtract 3y from both sides if we want, but that doesn’t help us, because the 3y just ends up on the other side.
Instead, we need to figure out how to cancel out the *y* on the left without introducing a *y* on the right. At first, this might seem impossible if we’re doing the same thing on both sides. But the trick is to remember that just because we’re doing the same thing to both sides doesn’t mean we have to **write** it the same way.
What if we took our first equation, -2x + 3y = 7, and subtracted 3y from the left side and 9 from the right side? We wouldn’t normally be allowed to do this, but because our other equation tells us that 3y = 9, *in this case* we’d actually be doing the same thing to both sides, just written differently.
Then we’d get -2x + 3y – 3y = 7 – 9. Now the 3y’s on the left side cancel out, and we’re left with -2x = -2, which is an equation we know how to solve.
For shorthand, we often say we “subtracted the second equation from the first”, and write it as:
-2x + 3y = 7
-( 3y = 9)
-2x = -2
but what we’re really doing is just subtracting the same value, written in two different ways, from both sides of the first equation.
Following so far?
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