Question

the perimiter of a rectangle is 96 ft. The ratio of its lenth to its with is 7:5. what is the permensions of rectangle?

Answer 1

The question gives you two equations.

Perimeter = 2*length + 2*width
Ratio of length to width = 7/5

We can rewrite these as:

96 = 2L + 2W
L/W = 7/5.

You can't solve an equation of two variables like you would solve an equation of one variable, but given TWO equations of two variables, you can use one of them to transform the other one into an equation of one variable. I will show you how to do this:

First I will solve for length in terms of width in one of my equations:
L = (7/5)W

And substitute this into my other equation:
96 = 2*(7/5)*W + 2W = (14/5)*W + 2*W = (24/5)W.

And just like you want, there is an equation of one variable which you can solve:
96 = (24/5)*W, so
W = 20.

You can now take this value and substitute it into either of your equations to get ANOTHER equation of just one variable (only this time for the OTHER variable). Either equation will work, but I'll use: 96 = 2L + 2W.

96 = 2*L + 2*20 = 2*L + 40.
56 = 2*L
L = 28.

And there you go, you have W = 20, L = 28. You can substitute these into either equation to check that it works out.

*NOTE* The key concept to take away from this is the idea of using one equation to reduce the number of variables in another equation. This concept is the basis of an entire field of study in mathematics called "linear algebra".