10-4 exercises

solve each equation or formula for the variable specified. assume that no variables are equal to zero.

1. $a = lw$ for $l$
2. $c = \pi d$ for $d$
3. $v = \pi r^2 h$ for $r$
4. $m = \frac{a + b}{2}$ for $b$
5. $p = i \times e$ for $i$
6. $p = \frac{e^2}{r}$ for $e$
7. $x = ay + t$ for $y$
8. $e = mc^2$ for $m$
9. $c = \frac{q}{v}$ for $v$
10. $s = 6s^2$ for $s$
11. $p = i^2 r$ for $i$
12. $x_c = \frac{1}{2\pi f c}$ for $c$
13. $f = \frac{1}{2\pi \sqrt{lc}}$ for $l$
14. $z = \sqrt{x^2 + r^2}$ for $x$

Question

10-4 exercises

solve each equation or formula for the variable specified. assume that no variables are equal to zero.

1. $a = lw$ for $l$
2. $c = \pi d$ for $d$
3. $v = \pi r^2 h$ for $r$
4. $m = \frac{a + b}{2}$ for $b$
5. $p = i \times e$ for $i$
6. $p = \frac{e^2}{r}$ for $e$
7. $x = ay + t$ for $y$
8. $e = mc^2$ for $m$
9. $c = \frac{q}{v}$ for $v$
10. $s = 6s^2$ for $s$
11. $p = i^2 r$ for $i$
12. $x_c = \frac{1}{2\pi f c}$ for $c$
13. $f = \frac{1}{2\pi \sqrt{lc}}$ for $l$
14. $z = \sqrt{x^2 + r^2}$ for $x$

Accepted Answer

{ "quality": "clear", "question_count": 14, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Literal Equations", "Formula Rearrangement" ], "new_concepts": [], "current_concepts": [ "Literal Equations", "Formula Rearrangement" ] } ## Solve linear literal equations Using the *Literal Equations* and *Formula Rearrangement* knowledge points $$ \begin{aligned} &\text{For 1: } A = lw \implies l = \frac{A}{w}\ &\text{For 2: } C = \pi d \implies d = \frac{C}{\pi}\ &\text{For 4: } m = \frac{a+b}{2} \implies 2m = a+b \implies b = 2m - a\ &\text{For 5: } P = I \times E \implies I = \frac{P}{E}\ &\text{For 7: } x = ay + t \implies x - t = ay \implies y = \frac{x-t}{a}\ &\text{For 8: } E = mc^2 \implies m = \frac{E}{c^2}\ &\text{For 9: } C = \frac{q}{V} \implies CV = q \implies V = \frac{q}{C} \end{aligned} $$ ## Solve quadratic and radical literal equations Using the *Literal Equations* and *Formula Rearrangement* knowledge points $$ \begin{aligned} &\text{For 3: } V = \pi r^2 h \implies r^2 = \frac{V}{\pi h} \implies r = \pm\sqrt{\frac{V}{\pi h}}\ &\text{For 6: } P = \frac{E^2}{R} \implies E^2 = PR \implies E = \pm\sqrt{PR}\ &\text{For 10: } S = 6s^2 \implies s^2 = \frac{S}{6} \implies s = \pm\sqrt{\frac{S}{6}}\ &\text{For 11: } P = I^2 R \implies I^2 = \frac{P}{R} \implies I = \pm\sqrt{\frac{P}{R}}\ &\text{For 14: } Z = \sqrt{X^2 + R^2} \implies Z^2 = X^2 + R^2 \implies X^2 = Z^2 - R^2 \implies X = \pm\sqrt{Z^2 - R^2} \end{aligned} $$ ## Solve rational and radical literal equations for denominators Using the *Literal Equations* and *Formula Rearrangement* knowledge points $$ \begin{aligned} &\text{For 12: } X_C = \frac{1}{2\pi f C} \implies C = \frac{1}{2\pi f X_C}\ &\text{For 13: } f = \frac{1}{2\pi\sqrt{LC}} \implies f^2 = \frac{1}{4\pi^2 LC} \implies L = \frac{1}{4\pi^2 f^2 C} \end{aligned} $$ | No. | Answer | |---|---| | 1 | $l = \frac{A}{w}$ | | 2 | $d = \frac{C}{\pi}$ | | 3 | $r = \pm\sqrt{\frac{V}{\pi h}}$ | | 4 | $b = 2m - a$ | | 5 | $I = \frac{P}{E}$ | | 6 | $E = \pm\sqrt{PR}$ | | 7 | $y = \frac{x-t}{a}$ | | 8 | $m = \frac{E}{c^2}$ | | 9 | $V = \frac{q}{C}$ | | 10 | $s = \pm\sqrt{\frac{S}{6}}$ | | 11 | $I = \pm\sqrt{\frac{P}{R}}$ | | 12 | $C = \frac{1}{2\pi f X_C}$ | | 13 | $L = \frac{1}{4\pi^2 f^2 C}$ | | 14 | $X = \pm\sqrt{Z^2 - R^2}$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Formula Rearrangement" ] }

No.	Answer
2	\(d = \frac{C}{\pi}\)
3	\(r = \pm\sqrt{\frac{V}{\pi h}}\)
4	\(b = 2m - a\)
5	\(I = \frac{P}{E}\)
6	\(E = \pm\sqrt{PR}\)
7	\(y = \frac{x-t}{a}\)
8	\(m = \frac{E}{c^2}\)
9	\(V = \frac{q}{C}\)
10	\(s = \pm\sqrt{\frac{S}{6}}\)
11	\(I = \pm\sqrt{\frac{P}{R}}\)
12	\(C = \frac{1}{2\pi f X_C}\)
13	\(L = \frac{1}{4\pi^2 f^2 C}\)
14	\(X = \pm\sqrt{Z^2 - R^2}\)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Solve linear literal equations

Solve quadratic and radical literal equations

Solve rational and radical literal equations for denominators

Answer: