Algebra equation - “in terms of” & variables, explanation required

I have a question in my book at the complicated equations chapter, it explain the difference between a term and a variable.

I would like to put here one question in order to somebody nice help me understand what is going on with it.

(I have actually cheated and saw the answers at the book, without actually knowing the answers, so I post here the solutions too).

This is the question on the book: "Match each equation to a description of which variable is in terms of which other variable."

$$T = 15d - 45 + 2^2.\qquad\qquad\text{Answer: Variable }T\text{ in terms of variable }d.$$

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What is the question asked in the book? There are probably some general instructions about what you are supposed to do with the equation you posted, but you did not include them. Please provide them so that what you post is understandable. –  Arturo Magidin Nov 7 '11 at 20:25

The equation you write gives you a relationship between the variable $T$ and the variable $d$. Each value of $d$ will give you a value of $T$ that makes the equation true, each value of $T$ will give you a value of $d$ that makes the equation true.

The equation gives you what is called an "explicit" expression of $T$ in terms of $d$: it tells you how to obtain the value of $T$ that corresponds to any particular value of $d$. That means that it "expresses [the value of] $T$ in terms of [the value of] $d$."

If you happen to know the value of $d$, then simply plugging in and performing the computations will give you the corresponding value of $T$. For example, if $d=4$, then plugging that into the right hand side of the equation gives you $$T = 15(4) - 45 + 2^2 = 60 - 45 + 4 = 19,$$ so when $d=4$, the value of $T$ that makes the equation true is $T=19$.

If you happen to know the value of $T$ instead, plugging it will not immediately yield a value for $d$; instead, you would need to do some algebra. If you happened to know that $T=19$, then you would have $$19 = 15d - 45 + 2^2.$$ From here, you would need to "solve for $d$" by performing algebraic manipulations to find that $d=4$.

So there is a subbstantial difference between using this equation to figure out the value of $T$ if you know the value of $d$; and using this equation to figure out the value of $d$ if you know the value of $T$. The value of $t$ is given explicitly as an expression involving the value of $d$; the value of $d$ is only given "implicitly" in terms of the value of $T$.

So we way that the variable $T$ is given "in terms" of the variable $d$: the expression tells you what to do to the value of $d$ in order to obtain the value of $T$.

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