Question 1060021
We need to write a system of equations here.


One equation will include the formula for the area/
The formula for the area of a rectangle is
Lw = a

where

L = length
W = width
a = area

Plug in the area where it is necessary.

Lw = 70

One equation is finished!

The next equation will be the situation described in your response.

The length is 3 inches added to the width.

In an equation, it is

3 + w = L

Now use your systems,

3 + w = L
Lw = 70

to solve for the length and width.




I'm going to solve using substitution.

Substitute 3 + w in for "L" in the second equation.


(3 + w)(w)=70

Distribute.

{{{3w + w^2 = 70}}}

Now we have a quadratic equation on our hands. Let's get this into standard form.


Subtract 70 from both sides.


{{{3w + w^2 - 70 = 0}}}

We cannot factor this, so solve using the quadratic formula.

{{{w = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

a = 3
b = 1
c = -70


Plug in and solve for w.


{{{w = (-1 +- sqrt( 1^2-4*3*-70 ))/(2*3) }}}

Simplify. Please view my steps.


{{{w = (-1 +- sqrt( 1+840 ))/(6) }}}

I squared 1 and multiplied -4 by 3, and then by -70.
I multiplied 2 times 3. 

{{{w = (-1 +- sqrt(841))/(6) }}}

I added one to 840.

{{{w = (-1 +- 29))/(6) }}}

I found the square root of 841, which is 29.


Now, we need to solve for "w". -1 can be ADDED or SUBTRACTED from 29, so we need to solve for both


Addition:

{{{w = (-1 + 29))/(6) }}}

-1 + 29 is 28.
28 / 6 can be simplified to 14/3.

w = 14/3



Subtraction:

{{{w = (-1 - 29))/(6) }}}

-1 - 29 is -30.
-30 / 6 is -5.

w = -5


Now we have a problem. w can equal -5 or 14/3.
-5 is the more logical answer since 14/3 will turn out to be a repeating decimal.

Our width is -5.


Don't worry, our system is not old news yet!
Let's return to it.

Lw = 70
w + 3 = L
Let's plug in -5 for "w" in the second equation and solve for "L".

-5 + 3 = L

-5 + 3 is -2.

-2 = L

YAY! We are finished!
Our width is -5 and our length is -2. These are our DIMENSIONS.