The width of a rectangle is the length minus 7 units. The area of the rectangle is 8 units. What is the length, in units, of the rectangle?

We can solve this problem using a system of equations. In this system of equations, let's have X = the length and Y = the width of the rectangle in question. Given this and the information above, we can write out the following two relationships:  

Equation 1: X * Y = 8 units

Equation 2: X - 7 = Y

We can draw these two relationships based upon the area data given for the rectangle and the relationship between the width and length. Plugging the value of Y from equation 2 into equation 1, we can isolate one variable, X, and solve for it:  

X * (X - 7) = 8.   Substituted equation 1

X^2 - 7X = 8 After being multiplied-out  

0 = X^2 - 7X - 8 After moving all terms to one side of the equation

0 = (X - 8) * (X + 1) After factoring the equation

X = 8, -1 Solve for X by finding an X that would make the equation true (equal to 0)

Now that we have solved for X, we have to possibilities for what X could be, however a length cannot be negative and this tells us that it must be 8 units, the other option. Plugging the value for X back into the original equation 2, we solve for Y:  

Equation 2: X - 7 = Y → 8 - 7 = Y = 1

Now that we see that Y = 1 units, we can check our solution by plugging it back into the area equation established earlier:  

Equation 1: X * Y = 8 units

Because X = 8 and Y = 1, this equation is satisfied and we know that we have the correct result.  

Final width = 1 unit, length = 8 units

Step-by-step explanation:

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