# Video: Solving Mathematical Equations Using the Arithmetic Mean

The arithmetic mean between −45 and 𝑚 is 24. Find 𝑚.

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### Video Transcript

The arithmetic mean between negative 45 and 𝑚 is 24. Find 𝑚.

We begin by recalling what we actually mean by the arithmetic mean. It’s the value we get from adding all of the values in some list and dividing by the total number of values we have. In this question, we could see we have just two values: negative 45 and. 𝑚. When we add these and divide by two cause there’re two numbers, we obtain 24. So let’s set this up as an equation.

The sum of negative 45 and 𝑚 is negative 45 plus 𝑚. Then, we can show that we’re dividing by two by using a fraction sign. So the arithmetic mean between negative 45 and 𝑚 is negative 45 plus 𝑚 all over two. But of course, we know that this is equal to 24. We now have an equation in 𝑚. And we can solve this for 𝑚.

To begin, we want to get rid of this fraction. And the inverse operation to dividing by two is multiplying by two. Remembering that whatever we do to one side of the equation, we have to do to the other, we’re going to multiply both sides of our equation by two. On the left, we’ve negative 45 plus 𝑚. Now, a common misconception here is to think that we need to multiply all of this by two. In fact, though, multiplying by two simply cancels out the divides by two so that negative 45 plus 𝑚 stays as it is. When we multiply 24 by two, we get 48.

Our ultimate aim is to get 𝑚 by itself. So we’re going to add 45 to both sides of this equation. Negative 45 plus 45 is zero. So we’re simply left with 𝑚 on the left-hand side. And on the right-hand side, we get 48 plus 45, which is equal to 93. And so, we found the value of 𝑚. It’s 93.