Transposition of formula

Consider the formula $A = \pi r^2$. Make $r$ the subject.

Solution: To make $r$ the subject, we first divide both sides by $\pi$:

Now, to isolate $r$, take the square root of both sides:

So, the formula for $r$ is $r = \sqrt{\frac{A}{\pi}}$.

Given the formula $v = \frac{d}{t}$, solve for $t$.

Solution: To solve for $t$, multiply both sides of the equation by $t$ to get rid of the denominator:

Now divide both sides by $v$ to isolate $t$:

So, the formula for $t$ is $t = \frac{d}{v}$.

In many cases, you will need to transpose formulas that involve square roots or squares. Here's how you can handle these situations:

Given the formula $A = \frac{1}{2} b h$, solve for $h$.

Solution: To solve for $h$, multiply both sides of the equation by 2:

Now, divide both sides by $b$ to isolate $h$:

So, the formula for $h$ is $h = \frac{2A}{b}$.

Consider the formula $x^2 + y^2 = r^2$. Make $y$ the subject.

Solution: To isolate $y$, first subtract $x^2$ from both sides:

Then, take the square root of both sides:

So, the formula for $y$ is $y = \pm \sqrt{r^2 - x^2}$.

Given the formula $F = \frac{1}{2} m v^2$, solve for $v$.

Solution: To solve for $v$, first multiply both sides of the equation by 2:

Now, divide both sides by $m$ to isolate $v^2$:

Next, take the square root of both sides:

So, the formula for $v$ is $v = \sqrt{\frac{2F}{m}}$.

When transposing formulas, follow these general steps:

1. Identify the symbol you want to make the subject of the formula.
2. Move all other terms to the opposite side using the inverse operations (addition/subtraction, multiplication/division, etc.).
3. If there are powers or roots involved, apply inverse operations like square roots or cubes to simplify.
4. Always check if the final formula makes sense with the given variables.

Given the formula for the area of a triangle $A = \frac{1}{2} b h$, solve for $h$ when $A = 12$ and $b = 6$.

Solution: To solve for $h$, multiply both sides by 2:

Substitute $A = 12$ and $b = 6$:

Now, divide both sides by 6:

So, $h = 4$.

In formulas with more than one variable, the same principles apply. Here's an example:

Given the formula $\frac{1}{x} + y = z$, solve for $x$.

Solution: To solve for $x$, subtract $y$ from both sides:

Now, take the reciprocal of both sides:

So, the formula for $x$ is $x = \frac{1}{z - y}$.