Question

What’s the algebraic expression for “32 divided by the quantity y plus 12”?

Answer 1

Hint: First understand the definition of an algebraic expression using examples. Now, assume the required algebraic expression as E. Take the sum of y and 12 and place it inside the bracket. In the next step consider 32 as the dividend and the above obtained sum as the divisor and divide 32 with it. Use the fact that the denominator of a fraction cannot be equal to 0 and form the expression E with the required condition.

Complete step-by-step solution:
Here we have been provided with the sentence “32 divided by the quantity y plus 12” and we are asked to convert it into an algebraic expression. But first we need to understand the term ‘algebraic expression’.
In mathematics, an algebraic expression is an expression that contains constants, variables and algebraic operations like: - addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number. For example: - \[5{{y}^{3}}z-3{{x}^{2}}y+xyz+3\] is an algebraic expression.
Let us come to the question. We have to write the algebraic expression for the sentence: - 32 divided by the quantity y plus 12. Let as assume the expression as E.
Now, here 32 is divided with the quantity y plus 12, it means y plus 12 is being considered as a single unit. So first considering the sum and placing it inside the bracket we get,
\[\Rightarrow \] (y + 12)
In the next step we have to perform the division in which 32 will be the dividend and the above sum will be the divisor. Therefore we have to write 32 in the numerator and (y + 12) in the denominator, so we get,
\[\Rightarrow E=\dfrac{32}{\left( y+12 \right)}\]
Now, for a dfraction to be defined the value of its denominator must not be equal to 0, so we have the condition: -
$\begin{align}
  & \Rightarrow y+12\ne 0 \\
 & \Rightarrow y\ne -12 \\
\end{align}$
Therefore, the required expression with the necessary condition is given as: -
\[\therefore E=\dfrac{32}{\left( y+12 \right)}\], where $y\ne -12$
Hence, the above algebraic expression is our answer.

Note: One may note that we must not substitute the obtained algebraic expression equal to any numerical value. This is because if we do so then the algebraic expression will become an algebraic equation. Always mention the condition where a function will become undefined because this is an important point regarding the continuity and differentiability of the function at that point in the calculus.