Rearranging Formulas to Highlight Quantities - Algebra 2

1. What is $x$ when you solve $y = \log_b(x)$ for $x$ (assume $b>0$, $b\ne 1$)?

   $x = b^y$. Apply the definition of logarithms: $\log_b(x) = y$ means $b^y = x$.

2. What is $P$ when you solve $A = P(1 + rt)$ for $P$ (assume $1+rt \ne 0$)?

   $P = \frac{A}{1 + rt}$. Divide both sides by $(1 + rt)$.

3. What is $x$ when you solve $y = (ax + b)^2$ for $x$ (with $a \ne 0$)?

   $x = \frac{\pm\sqrt{y} - b}{a}$. Take square root of both sides, then solve for $x$.

4. What is $V$ when you solve Ohm's law $V = IR$ for $V$?

   $V = IR$. No rearrangement needed; $V$ is already isolated.

5. What is $a$ when you solve $y = a(b^x)$ for $a$ (assume $b^x \ne 0$)?

   $a = \frac{y}{b^x}$. Divide both sides by $b^x$.

6. What is $b$ when you solve $y = \log_b(x)$ for $b$ (assume $x>0$, $y \ne 0$)?

   $b = x^{\frac{1}{y}}$. Rewrite as $b^y = x$, then solve for $b$.

7. What is $W$ when you solve $P = 2L + 2W$ for $W$ (treat $P,L$ as constants)?

   $W = \frac{P - 2L}{2}$. Subtract $2L$, then divide by $2$.

8. What is $x$ when you solve $y = \ln(x)$ for $x$?

   $x = e^y$. Apply the inverse relationship between $\ln$ and $e$.

9. What is $h$ when you solve $y = a(x - h)^2 + k$ for $h$ (assume $a \ne 0$)?

   $h = x \pm \sqrt{\frac{y - k}{a}}$. Isolate the squared term, take square root, then solve for $h$.

10. What is $x$ when you solve $y = a(x - h)^2 + k$ for $x$ (with $a \ne 0$)?

    $x = h \pm \sqrt{\frac{y - k}{a}}$. Isolate the squared term, take square root, then add $h$.

11. What is $P$ when you solve $A = P\left(1 + \frac{r}{n}\right)^{nt}$ for $P$?

    $P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$. Divide by the compound interest factor.

12. What is $r_2$ when you solve $A = \pi(r_1^2 - r_2^2)$ for $r_2$ (assume $r_2 \ge 0$)?

    $r_2 = \sqrt{r_1^2 - \frac{A}{\pi}}$. Solve for $r_2^2$ and take the square root.

13. What is $x$ when you solve $\frac{1}{x} = a$ (with $a \ne 0$) for $x$?

    $x = \frac{1}{a}$. Take the reciprocal of both sides.

14. What is $x$ when you solve $y = \sqrt{x - h}$ for $x$?

    $x = y^2 + h$. Square both sides, then add $h$.

15. What is $t$ when you solve $A = P(1 + rt)$ for $t$ (with $Pr \ne 0$)?

    $t = \frac{\frac{A}{P} - 1}{r}$. Divide by $P$, subtract $1$, then divide by $r$.

16. What is $m$ when you solve $y = mx + b$ (with $x \neq 0$) for $m$?

    $m = \frac{y - b}{x}$. Subtract $b$, then divide by $x$.

17. What is $x$ when you solve $ax + b = c$ (with $a \ne 0$) for $x$?

    $x = \frac{c - b}{a}$. Divide both sides by $a$ after isolating $ax$.

18. What is $b$ when you solve triangle area $A = \frac{1}{2}bh$ for $b$ (with $h \ne 0$)?

    $b = \frac{2A}{h}$. Multiply by $2$, then divide by $h$.

19. What is $x$ when you solve $y = \frac{a}{x} + b$ for $x$ (with $y \ne b$)?

    $x = \frac{a}{y - b}$. Subtract $b$, then divide $a$ by the result.

20. What is $r$ when you solve $d = rt$ for $r$ (with $t \ne 0$)?

    $r = \frac{d}{t}$. Divide both sides by $t$.

21. What is $b$ when you solve $y = mx + b$ for $b$?

    $b = y - mx$. Subtract $mx$ from both sides.

22. What is $R$ when you solve Ohm's law $V = IR$ for $R$ (with $I \ne 0$)?

    $R = \frac{V}{I}$. Divide both sides by $I$ to isolate resistance.

23. What is $x$ when you solve $ax^2 = b$ for $x$ (with $a \ne 0$)?

    $x = \pm\sqrt{\frac{b}{a}}$. Divide by $a$, then take square root with $\pm$.

24. What is $x$ when you solve $a(x + h) = k$ (with $a \ne 0$) for $x$?

    $x = \frac{k}{a} - h$. Divide by $a$, then subtract $h$ from both sides.

25. What is $I$ when you solve Ohm's law $V = IR$ for $I$ (with $R \ne 0$)?

    $I = \frac{V}{R}$. Divide both sides by $R$ to isolate current.

26. What is $b$ when you solve $y = a(b^x)$ for $b$ (assume $a>0$, $y>0$)?

    $b = \left(\frac{y}{a}\right)^{\frac{1}{x}}$. Take the $x$th root of $\frac{y}{a}$.

27. What is $h$ when you solve triangle area $A = \frac{1}{2}bh$ for $h$ (with $b \ne 0$)?

    $h = \frac{2A}{b}$. Multiply by $2$, then divide by $b$.

28. What is the correct first step to isolate $x$ in $ax + b = c$ (with $a \ne 0$)?

    Subtract $b$: $ax = c - b$. Use inverse operations to isolate the term with $x$.

29. What is $x$ when you solve $\frac{x}{a} = b$ (with $a \ne 0$) for $x$?

    $x = ab$. Multiply both sides by $a$ to eliminate the fraction.

30. What is $x$ when you solve $a = \frac{b}{x}$ (with $a \ne 0$) for $x$?

    $x = \frac{b}{a}$. Cross-multiply to get $ax = b$, then divide by $a$.

31. What is $r$ when you solve $A = P(1 + rt)$ for $r$ (with $Pt \ne 0$)?

    $r = \frac{\frac{A}{P} - 1}{t}$. Divide by $P$, subtract $1$, then divide by $t$.

32. What is $x$ when you solve $y = \sqrt{ax + b}$ for $x$ (with $a \ne 0$)?

    $x = \frac{y^2 - b}{a}$. Square both sides, subtract $b$, then divide by $a$.

33. What is $k$ when you solve $y = a(x - h)^2 + k$ for $k$?

    $k = y - a(x - h)^2$. Subtract the squared term from both sides.

34. What is $r$ when you solve $V = \pi r^2 h$ for $r$ (assume $h>0$)?

    $r = \sqrt{\frac{V}{\pi h}}$. Divide by $\pi h$, then take the square root.

35. What is $x$ when you solve $y = \frac{ax + b}{c}$ for $x$ (with $a \ne 0$, $c \ne 0$)?

    $x = \frac{cy - b}{a}$. Multiply by $c$, subtract $b$, then divide by $a$.

36. What is $c$ when you solve the Pythagorean theorem $a^2 + b^2 = c^2$ for $c$?

    $c = \sqrt{a^2 + b^2}$. Take the square root of both sides.

37. What is $x$ when you solve $a(x - h) = k$ (with $a \ne 0$) for $x$?

    $x = \frac{k}{a} + h$. Divide by $a$, then add $h$ to both sides.

38. What is $x$ when you solve $ax = b$ (with $a \ne 0$) for $x$?

    $x = \frac{b}{a}$. Divide both sides by $a$ to isolate $x$.

39. What is $h$ when you solve the cylinder volume formula $V = \pi r^2 h$ for $h$?

    $h = \frac{V}{\pi r^2}$. Divide both sides by $\pi r^2$.

40. What is $r$ when you solve the circle area formula $A = \pi r^2$ for $r$?

    $r = \sqrt{\frac{A}{\pi}}$. Divide by $\pi$, then take the square root.

41. What is $r$ when you solve circumference $C = 2\pi r$ for $r$?

    $r = \frac{C}{2\pi}$. Divide both sides by $2\pi$.

42. What is $x$ when you solve $y = mx + b$ (with $m \ne 0$) for $x$?

    $x = \frac{y - b}{m}$. Subtract $b$, then divide by $m$.

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