Can someone help me with this calculus question?

Question

i dont get it.. its multiple choice btw

x=2-t

y=t t²

the above pair of parametric equations is equivalent to which of the following equations in x and y

a) f y=2 t²

b) y=(x-2)(x-3)

c) y=2 x²

d) y=(x 2)(x 3)

Answer 1

Basically the question is asking.... "how can you rewrite the given two equations.... as one single equation alone (incorporating the two equations together....)???

Given: "x = 2 - t" and y = "t + t²"

What you have to do: Combine the two equations and right them as one equation .... as "y = [something]"

Okay.... so take the "x = 2 - t" equation... and solve for "t" .... (that is, rewrite the equation so that "t" is on the left side....)

x = 2 - t

Add "t" to both sides... like this...

x = 2 - t

+t ... +t

-------------

x+t = 2 + 0

x + t = 2

Now.... subtract "x" from both sides.... like this...

x + t = 2

-x ......... -x

-------------------

0 + t = 2 - x

t = 2 - x

Now that you solved for "t".... plug in "2-x" for "t" in the 2nd equation.....

So.... your y = t + t² equation...... becomes.....

y = (2-x) + (2-x)²

Now you just have to expand the right side and simplify... like this....

y = (2-x) + (2-x)²

.. = 2 - x + (2-x)(2-x) <===== use FOIL method to expand (2-x)(2-x)

.. = 2 - x + (2)(2) + (2)(-x) + (-x)(2) + (-x)(-x)

.. = 2 - x + 4 - 2x -2x + x²

.. = 2 - x + 4 - 4x + x²

.. = 6 - 5x + x²

.. = x² - 5x + 6

(Note: to factor the polynomial x² -5x + 6... you need two numbers that sum up to -5 and whose product is also 6)

... = (x-2)(x-3)

So.... the pair of parametric equations is equivalent to B) y = (x-2)(x-3)