Example 1: Simple Linear Equation

Make \( x \) the subject of the formula:
\[
y = 3x + 5
\]

Step 1: Subtract 5 from both sides:
\[
y – 5 = 3x
\]

Step 2: Divide both sides by 3 to isolate \( x \):
\[
x = \frac{y – 5}{3}
\]

Example 2: Formula with Fractions

Make \( a \) the subject of the formula:
\[
\frac{p}{q} = a + b
\]

Step 1: Subtract \( b \) from both sides:
\[
\frac{p}{q} – b = a
\]

Step 2: Simplify (if necessary):
\[
a = \frac{p}{q} – b
\]

Example 3: Formula with Multiplication and Division

Make \( t \) the subject of the formula:
\[
s = ut + \frac{1}{2}at^2
\]

Step 1: Subtract \( ut \) from both sides:
\[
s – ut = \frac{1}{2}at^2
\]

Step 2: Multiply through by 2 to eliminate the fraction:
\[
2(s – ut) = at^2
\]

Step 3: Divide through by \( a \) to isolate \( t^2 \):
\[
t^2 = \frac{2(s – ut)}{a}
\]

Step 4: Take the square root:
\[
t = \pm \sqrt{\frac{2(s – ut)}{a}}
\]

Example 4: Formula with a Square Root

Make \( r \) the subject of the formula:
\[
A = \pi r^2
\]

Step 1: Divide both sides by \( \pi \):
\[
\frac{A}{\pi} = r^2
\]

Step 2: Take the square root of both sides:
\[
r = \sqrt{\frac{A}{\pi}}
\]

Example 5: Rearranging a Physics Equation

Make \( F \) the subject of the formula:
\[
P = \frac{F}{A}
\]

Step 1: Multiply both sides by \( A \) to isolate \( F \):
\[
F = P \cdot A
\]

Recap of Techniques to Solve Transposition Formula (1-5)

1. Addition/Subtraction: Move terms across the equation by adding or subtracting.
2. Multiplication/Division: Eliminate coefficients or fractions by multiplying or dividing.
3. Square Roots: Take the square root to solve for squared variables.
4. Clear Fractions: Multiply through by denominators to simplify equations.

Example 6: Rearranging a Formula with a Logarithm

Make \( x \) the subject of the formula:
\[
y = \ln(x) + k
\]

Step 1: Subtract \( k \) from both sides:
\[
y – k = \ln(x)
\]

Step 2: Exponentiate both sides (use \( e \) as the base):
\[
x = e^{y – k}
\]

Example 7: Rearranging a Formula with an Exponent

Make \( t \) the subject of the formula:
\[
P = P_0 e^{kt}
\]

Step 1: Divide both sides by \( P_0 \):
\[
\frac{P}{P_0} = e^{kt}
\]

Step 2: Take the natural logarithm of both sides:
\[
\ln\left(\frac{P}{P_0}\right) = kt
\]

Step 3: Divide by \( k \) to isolate \( t \):
\[
t = \frac{\ln\left(\frac{P}{P_0}\right)}{k}
\]

Example 8: Rearranging a Formula with a Square Root

Make \( h \) the subject of the formula:
\[
T = 2\pi \sqrt{\frac{h}{g}}
\]

Step 1: Divide both sides by \( 2\pi \):
\[
\frac{T}{2\pi} = \sqrt{\frac{h}{g}}
\]

Step 2: Square both sides to remove the square root:
\[
\left(\frac{T}{2\pi}\right)^2 = \frac{h}{g}
\]

Step 3: Multiply through by \( g \):
\[
h = g \left(\frac{T}{2\pi}\right)^2
\]

Example 9: Rearranging a Formula with a Product

Make \( x \) the subject of the formula:
\[
z = \frac{y}{x + b}
\]

Step 1: Multiply through by \( x + b \) to eliminate the fraction:
\[
z(x + b) = y
\]

Step 2: Expand the left-hand side:
\[
zx + zb = y
\]

Step 3: Subtract \( zb \) from both sides:
\[
zx = y – zb
\]

Step 4: Divide through by \( z \) to isolate \( x \):
\[
x = \frac{y – zb}{z}
\]

Example 10: Rearranging a Quadratic Formula

Make \( v \) the subject of the formula:
\[
s = ut + \frac{1}{2}at^2
\]

Step 1: Subtract \( ut \) from both sides:
\[
s – ut = \frac{1}{2}at^2
\]

Step 2: Multiply through by 2:
\[
2(s – ut) = at^2
\]

Step 3: Divide through by \( a \):
\[
t^2 = \frac{2(s – ut)}{a}
\]

Step 4: Take the square root:
\[
t = \pm \sqrt{\frac{2(s – ut)}{a}}
\]

Techniques Recap on Transposition

1. Fractions: Clear denominators by multiplying.
2. Exponents and Logs: Use exponentiation or logarithms to isolate variables.
3. Square Roots: Square or root appropriately.
4. Quadratic Forms: Solve using known methods like factorization or completing the square.
5. Logical Steps: Always do the opposite operation to isolate the desired variable.