Changing the Subject of a Formula for ESAT

The Fundamental Rules of Rearranging

Changing the subject of a formula requires expressing one variable in terms of the other variables. To achieve this, the formula must be rearranged according to the standard rules of arithmetic and algebra. The goal is to perform operations that isolate the new subject. The following operations can be applied to both sides of a formula to maintain equality:

1. You can add or subtract the same quantity or term from each side.
2. You can multiply or divide both sides by the same non-zero quantity.
3. You can invert both sides of a formula if both sides consist of a single fraction with non-zero denominators.
4. You can square both sides or raise both sides to the same non-zero power.
5. You can take the square root of both sides.

Changing the Subject Using Subtraction

When the variable you wish to isolate appears in multiple terms or alongside other variables, you must first group the terms containing that variable. Consider the following example where we make $p$ the subject of the formula:

$q + 2p = p + 4q - 6r$

First, subtract $p$ from both sides to ensure all terms containing $p$ are on the same side: $q + 2p - p = 4q - 6r$

Collect the like terms on the left side: $q + p = 4q - 6r$

Next, subtract $q$ from both sides to isolate the $p$ term: $p = 4q - 6r - q$

Collect the like terms on the right side: $p = 3q - 6r$

Finally, factorise the expression by taking out the common factor of 3 to simplify the result: $p = 3(q - 2r)$

Changing the Subject Using the Four Functions

In more complex formulae involving brackets and fractions, it is often necessary to expand and multiply out denominators before isolating the subject. Consider making $x$ the subject of the following formula:

$y = 3(x - 2) + \frac{x}{2}$

First, multiply out the bracket on the right hand side: $y = 3x - 6 + \frac{x}{2}$

To eliminate the fraction, multiply both sides of the formula by 2: $2y = 6x - 12 + x$

Collect the like terms involving $x$: $2y = 7x - 12$

Add 12 to both sides to further isolate the $x$ term: $2y + 12 = 7x$

Finally, divide by 7 to make $x$ the subject: $x = \frac{2y + 12}{7}$

Variables in the Denominator of a Fraction

If the variable you wish to isolate is located in the denominator, you must move it to the numerator. This often involves finding a common denominator or inverting the fractions. For example, make $u$ the subject of this formula:

$\frac{1}{v} - \frac{1}{u} = \frac{1}{p}$

First, rearrange the formula to isolate the term containing $u$ on one side: $\frac{1}{v} - \frac{1}{p} = \frac{1}{u}$

Now, combine the terms on the left hand side by placing them over a common denominator, which is $vp$: $\frac{p - v}{vp} = \frac{1}{u}$

Since both sides are now single fractions, you can invert both fractions, essentially turning them both upside down: $\frac{vp}{p - v} = \frac{u}{1}$

This gives the final result: $u = \frac{vp}{p - v}$

Formulae Involving Square Roots

When a variable is trapped inside a square root, you must apply the inverse operation, which is squaring, to free it. Consider making $x$ the subject of the formula below:

$y = \sqrt{\frac{3x + 5}{4}}$

First, square both sides of the formula: $y^2 = \frac{3x + 5}{4}$

Now, rearrange the formula to isolate $x$. Multiply both sides by 4: $4y^2 = 3x + 5$

Subtract 5 from both sides: $4y^2 - 5 = 3x$

Finally, divide by 3 to isolate $x$ completely: $x = \frac{4y^2 - 5}{3}$