Systems of equations

Question

Here's the question.

The dimensions of the rectangular foundation of a house is represented by the expressions (x+5) and (x-4). The area of the foundation can be represented by y+20 and the perimeter by y-12.

Create a systems of equations to represent the area and perimeter of the home's foundation.

Then, algebraically solve to find the dimensions themselves.

Answer 1

Setting the equations up is the harder part usually. So let's see what you have for information.

You have the area and the perimeter. You know the formulas for area and perimeter of a rectangle, right?

A = L * W

P = 2L + 2W

And you have the length and width, in terms of X, plus the area and perimeter, in terms of Y. So you substitute into the two formulas:

Area: y + 20 = (x+5) * (x −4)

Perimeter: y − 12 = 2((x+5) + 2(x −4)

To solve, foil the first one out and solve for y. Distribute the second one, combine like terms, and solve for y. I'll let you do this... if you're in pre-calc, I think you can!

Now you have two equations, both with y isolated. If two different things both equal Y, then the two equal each other, right? For example, if you have:

y = x + 2 and y = x²−4 then x+2 = x²−4 and 0 = x² −x −6 ... which you can solve by factoring to get x = 3 or x = −2. Note that this is NOT your exact problem, just an example. You have to solve your problem algebraically for yourself. :-)

Once you get an answer for x, remember to double check that it is a valid answer. In Algebra problems, we can have negative answers, but you do NOT want a negative measurement in Geometry. It is likely that one of your roots will be invalidated for that reason, leaving you the other as correct.

Once you determine what your valid answer for X is, use it to find Y.

I hope this gives you the key to the problem... without handing you TOO much.