Problem: You want formulas and methods to change the subject of a formula, that is, to make a chosen variable the subject of an equation. 1. Linear equations. Example: isolate $x$ in $ax + b = c$. Step: subtract $b$ from both sides to get $ax = c - b$. Step: divide both sides by $a$ to obtain $x = $\frac{c - b}{a}$. Explanation: perform inverse operations in reverse order: undo addition, then undo multiplication. 2. Fractional equations and cross-multiplication. Example: isolate $x$ in $\frac{p}{x} + q = r$. Step: subtract $q$ to get $\frac{p}{x} = r - q$. Step: multiply both sides by $x$ and divide by $r - q$ (assuming $r - q \neq 0$) to get $x = $\frac{p}{r - q}$. Explanation: eliminate denominators by multiplying through and then isolate the variable. 3. Variable in denominator with two fractions. Example: solve $\frac{m}{n} = \frac{p}{x}$ for $x$. Step: cross-multiply to get $m x = n p$. Step: divide by $m$ to obtain $x = $\frac{n p}{m}$. 4. Variable in exponent. Example: solve $a^x = b$ for $x$ with $a>0$, $a \neq 1$. Step: take logarithms to get $x \ln a = \ln b$. Step: divide by $\ln a$ to obtain $x = $\frac{\ln b}{\ln a}$. Explanation: use logarithms to bring the exponent down. 5. Variable inside a logarithm. Example: solve $\log_a x = b$ for $x$. Step: exponentiate with base $a$ to get $x = a^b$. 6. Radical expressions. Example: solve $\sqrt{x} = y$ for $x$. Step: square both sides to get $x = y^2$. Note: check for extraneous roots when dealing with even roots and sign constraints. 7. Quadratic equations. Example: solve $a x^2 + b x + c = 0$ for $x$. Step: apply the quadratic formula to get $$x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$$. Brief derivation: complete the square on $a x^2 + b x + c = 0$ and solve for $x$. 8. Absolute value. Example: solve $|x| = a$ for $x$. Step: write two cases $x = a$ and $x = -a$. 9. Trigonometric functions. Example: solve $\sin x = y$ for $x$. Step: apply the inverse function to get $x = \arcsin y$ (plus periodic general solutions when required). 10. Systems and substitution. Tip: when the variable appears in multiple equations, solve one equation for the variable using the above methods and substitute into the other equation. 11. General strategy and tips. - Undo operations in reverse order of application. - Use algebraic identities and factorization when possible, e.g., factor common terms to isolate the variable. - Keep track of domain restrictions and avoid dividing by zero. - Check solutions by substituting back into the original equation. Summary of common formulas and operations: - Linear: $x = $\frac{c - b}{a}$ from $ax + b = c$. - Reciprocal/fraction: $x = $\frac{p}{r - q}$ from $\frac{p}{x} + q = r$. - Exponential: $x = $\frac{\ln b}{\ln a}$ from $a^x = b$. - Logarithm: $x = a^b$ from $\log_a x = b$. - Quadratic: $$x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$$. End of explanation.