How do i solve this system of equations?
y=(x+1)^2-2
and y-2x-3

Question

How do i solve this system of equations?
y=(x+1)^2-2
and y-2x-3

whpalmer4 · Answer

is the second equation $$y=2x-3$$?

whpalmer4 · Answer

Well, let's assume that it is.  

$$y = (x+1)^2-2$$$$y=2x-3$$

We can solve this by substitution.  Everywhere you see $y$ in the first equation, replace it with the right hand side of the second equation (which is also equal to $y$).  This gives us an equation all in terms of just the variable $x$:

$$2x-3 = (x+1)^2-2$$Now you can expand that out:
$$2x-3 = (x+1)(x+1)-2$$$$2x-3=x^2+1x+1x+1*1-2$$$$2x-3=x^2+2x-1$$and get all the terms on one side and you have a quadratic to solve.  You know how to do that?

cutiecomittee123 · Answer

The second equation is y-2x=3

whpalmer4 · Answer

Okay, can you solve $$y-2x=3$$ for $y$?

cutiecomittee123 · Answer

y=3+2x

whpalmer4 · Answer

Right.  So now we do the same thing I did, except with the correct equation.

$$y = (x+1)^2+2$$$$y = 2x+3$$

$$2x+3 = (x+1)^2 + 2$$
Expand both sides, then combine all the terms on one side and solve the quadratic.

cutiecomittee123 · Answer

wait it should be 3+2x=(x+1)^2-2

whpalmer4 · Answer

doesn't $$2x+3 = 3+2x$$?

whpalmer4 · Answer

sorry, I switched a sign, that's what you were pointing out, my bad!

whpalmer4 · Answer

$$3+2x = (x+1)^2-2$$

cutiecomittee123 · Answer

yes but the original equation is y=(x+1)^2-2 and if we are plugging in y , and y=2x+3

cutiecomittee123 · Answer

Yeah thats what I got :))

whpalmer4 · Answer

Glad one of us is paying close attention :-)

cutiecomittee123 · Answer

so now i get 2x+3=x^2+1+2

cutiecomittee123 · Answer

thats after expanding the equation

whpalmer4 · Answer

close...
$$(x+1)^2 = (x+1)(x+1) = x*x + 1*x + 1*x + 1*1 = x^2 + 2x + 1$$

so [2x+3 = x^2+2x+1 - 2\]$$2x+3 = x^2+2x-1$$
right?

cutiecomittee123 · Answer

yes got that

whpalmer4 · Answer

okay, get all the terms collected on one side, what do you get?

cutiecomittee123 · Answer

0=x^2-4

whpalmer4 · Answer

Okay, and what are the solutions to that equation?

whpalmer4 · Answer

That's a simple one to solve:
$$x^2-4 = 0$$$$x^2 = 4$$Take the square root of both sides and you see $$x=\pm 2$$
Have to be sure to include the negative square root or we won't get all of the answers!

Now we need to find the values of $y$ that go along with those values of $x$.  We can use either equation, so might as well use the simpler one: $$y=2x+3$$

Plug in both values of $x$ and get the corresponding values of $y$.

cutiecomittee123 · Answer

okay just a sec

cutiecomittee123 · Answer

I get (2,7) and (-2,-1)

whpalmer4 · Answer

So do I!

whpalmer4 · Answer

and we can easily test them:

$$2(2)+3 = (2+1)^2-2$$$$7 = 9-2\checkmark$$
$$2(-2)+3 = (-2+1)^2-2$$$$-1=1-2\checkmark$$

whpalmer4 · Answer

So we can do this any time we can rearrange the stuff in one equation to give us something we can substitute into the other one.

cutiecomittee123 · Answer

sweet thanks

whpalmer4 · Answer

Here's a tip that doesn't really matter here so much, but does in problems that you will encounter: always try your answers out in the original equation and make sure they work!

Whenever you have to square both sides of the equation, you can get what called "spurious" solutions which come out just the same way as the real ones do, but don't actually make the equations true when you plug the numbers in.  Rather than remembering exactly what kinds of problems require you to do this, I prefer to simply always check my answers!

anonymous · Answer

Cool! @whpalmer4