The length of a rectangle is 11 centimeters less than six times its width. Its area is 21 square centimeters. Find the dimensions of the rectangle.

The width is 3cm, and the length is 7cm.

Step-by-step explanation:

Let's call the length of the rectangle , the width of the rectangle , and the area of the rectangle , and write down everything we know about them.

The length of the rectangle is 11 centimetres less than six times its width, so we know that .

We also know that . But what exactly is the area of a rectangle? Well, it's the product of its length and width, i.e. .

Then we have two equations involving and :

, and .

So how do we find and ? Let's substitute our first equation into our second, like so:

We can rearrange this new equation to get everything on the same side, so that we have a standard quadratic equation, like this:

.

We can use the quadratic formula for this. You should have seen this before:

,

where are the coefficients of our quadratic equation. Plugging the values in, we see

.

With a little further cancellation, we can find our two possible values of are

 and .

Since is a width of a rectangle, it can't possibly be negative, and so it must be 3.

We can now plug in this value into our equation from earlier, namely . We can see that

,

and so we get our answer, cm, cm.

We can prove that this is right by multiply 3 and 7 together, and seeing that we get 21, the area we expect.

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