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This next problem is an Algebra problem with a difficulty rating of HARD. This question
is typical of problems that appear later on the ACT Math Test. Try to work this problem
in one minute and then I'll go through the solution.
The correct answer is G. We're given y as a function of t, and x as
a function of y. We want to know x as a function of t.
We can get this by combining these two equations. So let's start with the expression for x.
Now we want to get rid of the y in this expression, and introduce some t's.
Here's the other equation; y is equal to t minus 2. This is already in a form that allows
us to plug right in for the y in the expression for x.
So we take t-2 and replace the y anywhere it shows up in the expression for x.
So we have x is equal to, now instead of y squared we have (t-2) squared, plus 3 times,
and instead of y we put in a t-2, minus 1. Let's expand that first term, so we have t
squared minus 4t plus 4, and then we'll multiply the 3 through in the middle term. So we have
plus 3t-6, minus 1. Now we want to combine like terms. There's
only one term that has a t squared, so we'll put that out front.
And looking for terms that have a t in them, we see a negative 4t and plus 3t, so we'll
group those together. And then we'll group the constants, 4 minus
6 minus 1. That gives us t squared, and -4t+3t is -1t,
or just minus t. And then 4 minus 6 minus 1 is negative 3 so
we get minus 3. And this is our answer, which is choice G.
Here's a quick reminder about combining like terms in polynomials. This is one of those
fundamental skills that's likely to be needed on several of the problems on your ACT Math
Test. Like terms in a polynomial have the same variable
with the same power. Like terms can be combined by adding the real
part, that is, adding the constant in front of the variable.
For example, 4 x squared plus 2x minus x squared plus 3x. If we want to combine like terms
to simplify this expression, we would group the 4 x squared and the minus x squared and
get 3 x squared for the x squared part. And then we would group the 2x and the 3x and
get 5x. Our simplification would be 3 x squared plus 5x.