Making a Variable the Subject, foundations for calculus lesson

A thermometer app shows Celsius, but your oven dial is in Fahrenheit. The formula you have turns Celsius into Fahrenheit, and you need it the other way around. Rearranging a formula answers exactly that kind of question, and calculus asks it all the time, from implicit differentiation to related rates. The algebra is short. The care is in noticing when a step is allowed and which values still make sense.

Key terms

Subject of a formula

Definition A formula is an equation that relates several variables. A variable is the subject of the formula when it stands alone on one side and does not appear anywhere on the other side. To make a variable the subject means to replace the formula by an equation of the form

\begin{gathered} \text{variable} = \text{expression} \\[0.4em] \text{without that variable}, \end{gathered}

which has exactly the same solutions as the original formula for all the values allowed by the conditions you state along the way.

In plain words Decide which letter you want, then get it by itself on one side. The other side then tells you how to compute it from everything else.

Example The circumference of a circle is $C = 2\pi r$ , so $C$ is the subject. Dividing both sides by the nonzero number $2\pi$ gives

r = \frac{C}{2\pi}.

Now $r$ is the subject. A circle with circumference $C = 10\pi$ centimeters has radius $r = 5$ centimeters.

Parameter

Definition A parameter is a quantity that is held fixed while the other variables in a formula vary. Different values of the parameter give different members of a family of formulas or graphs. In this lesson every variable and parameter is a real number, and a parameter may come with conditions such as $m \ne 0$ or $L > 0$ that must be stated whenever a step depends on them.

In plain words A parameter is a setting you choose once and then leave alone. Change the setting and you get a different line, a different rectangle or a different circle.

Example In $y = mx + b$ the letters $x$ and $y$ are the variables, and $m$ and $b$ are parameters. Choosing $m = 2$ and $b = -1$ gives the single line $y = 2x - 1$ . Choosing $m = 2$ and letting $b$ run over all real numbers gives a family of parallel lines with slope $2$ .

Inverse operation

Definition An inverse operation undoes another one. Adding $c$ is undone by subtracting $c$ , and multiplying by a number $c \ne 0$ is undone by dividing by $c$ . Two moves always turn an equation into an equivalent one, with exactly the same solutions. The first is adding or subtracting the same real number on both sides, or the same expression as long as it is defined for every value under consideration. The second is multiplying or dividing both sides by the same quantity, provided that quantity is not zero.

To isolate a variable, undo the operations applied to it in the reverse of the order in which they were applied. Operations that are not one-to-one, such as squaring, have no inverse on all real numbers, so they need extra care.

In plain words Peel the variable free from the outside in. Whatever was done to it last, you undo first.

Example Fahrenheit is computed from Celsius by multiplying by $\frac{9}{5}$ and then adding $32$ ,

F = \frac{9}{5}C + 32.

Undo the last step first by subtracting $32$ , then multiply both sides by $\frac{5}{9}$ ,

C = \frac{5}{9}(F - 32).

A reading of $F = 212$ gives $C = \frac{5}{9} \cdot 180 = 100$ , the boiling point of water, as it should.

Nonzero condition

Definition Dividing both sides of an equation by a quantity $m$ produces an equivalent equation only when $m \ne 0$ . When you divide by an expression that contains a parameter, the rearranged formula carries the condition that this expression is nonzero. The case where it equals zero must be examined separately, directly from the original equation and without dividing.

In plain words You may never divide by zero, even when the zero is hiding inside a letter. Write the condition next to the answer, and then look at the forbidden case on its own.

Example Make $x$ the subject of the line equation $y = mx + b$ . Subtracting $b$ gives $y - b = mx$ . If $m \ne 0$ , divide by $m$ ,

x = \frac{y - b}{m}, \qquad m \ne 0.

For the line $y = 2x - 1$ and the height $y = 7$ this gives $x = \frac{7 + 1}{2} = 4$ , and indeed $2 \cdot 4 - 1 = 7$ .

If $m = 0$ the original equation reads $y = b$ , a horizontal line. A height $y \ne b$ is never reached, so no $x$ works. The height $y = b$ is reached at every $x$ , so there is no single answer. Either way $x$ cannot be made the subject, which is exactly what the condition $m \ne 0$ warned about.

Units

Definition In a formula about measured quantities, both sides of an equation must have the same units, and only quantities with the same units may be added or subtracted. Rearranging preserves this, so the units of the new subject can be found by doing the same operations to the units. A mismatch signals an algebra mistake.

In plain words The units follow the algebra. If the answer for a time comes out in meters, something went wrong.

Example Distance at constant speed is $d = vt$ . For a speed $v \ne 0$ , dividing by $v$ gives

t = \frac{d}{v}.

The units are kilometers divided by kilometers per hour, which is hours. Covering $d = 150$ kilometers at $v = 60$ kilometers per hour takes

t = \frac{150}{60} = 2.5 \text{ hours.}

Admissible value

Definition An admissible value of a variable or parameter is a value for which the formula is defined and still describes the intended situation. A rearranged formula may be algebraically valid for every real number while only some of those numbers are admissible. The admissible set must be stated with the formula, and it is carried over to the rearranged version in terms of the new subject.

In plain words The algebra does not know that a length cannot be negative. You have to tell it, and the restriction stays with the formula after you rearrange it.

Example A rectangle with length $L$ and width $W$ has perimeter $P = 2L + 2W$ . For an ordinary rectangle $L > 0$ and $W > 0$ , so $P$ is positive. To make $W$ the subject, subtract $2L$ and divide by the nonzero number $2$ ,

W = \frac{P - 2L}{2} = \frac{P}{2} - L.

The condition $W > 0$ now becomes $L < \frac{P}{2}$ , so the admissible lengths for a given perimeter are $0 < L < \frac{P}{2}$ . With $P = 20$ meters and $L = 7$ meters the width is $W = 10 - 7 = 3$ meters. With $L = 12$ meters the formula still returns a number, $W = -2$ , but no rectangle has that width, so $L = 12$ is not admissible.

The rearranged formula also reads well in context. For a fixed perimeter, each extra meter of length costs exactly one meter of width. That kind of tradeoff is the starting point of many optimization problems in calculus.

Check yourself

Make $y$ the subject of $3x - 2y = 6$ , and find $y$ when $x = 4$ .

Show the answer

Subtract $3x$ to get $-2y = 6 - 3x$ , then divide by the nonzero number $-2$ ,

y = \frac{3x - 6}{2} = \frac{3}{2}x - 3.

At $x = 4$ this gives $y = 3$ , and the check $3 \cdot 4 - 2 \cdot 3 = 6$ works.

An object moves with constant acceleration $a$ , so its velocity after time $t$ is $v = v_0 + at$ . Make $t$ the subject and say what happens when $a = 0$ .

Show the answer

Subtract $v_0$ and divide by $a$ , which is allowed only when $a \ne 0$ ,

t = \frac{v - v_0}{a}, \qquad a \ne 0.

When $a = 0$ the velocity is always $v_0$ . A different velocity is never reached, and the velocity $v_0$ is reached at every time, so $t$ cannot be found from $v$ .

A fence of total length $30$ meters encloses a rectangle, so $2L + 2W = 30$ . Make $L$ the subject and give the admissible widths.

Show the answer

Dividing by $2$ gives $L + W = 15$ , so $L = 15 - W$ . A real rectangle needs $W > 0$ and $L > 0$ , and $15 - W > 0$ means $W < 15$ . The admissible widths are $0 < W < 15$ meters.

Making a Variable the Subject

Key terms

Subject of a formula

Parameter

Inverse operation

Nonzero condition

Units

Admissible value

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus