Answer :
Certainly! Let's solve each equation for the indicated variable step-by-step.
### 13. [tex]\(\frac{b}{c} = a; \quad c\)[/tex]
To solve for [tex]\(c\)[/tex]:
[tex]\[ \frac{b}{c} = a \][/tex]
Multiply both sides by [tex]\(c\)[/tex]:
[tex]\[ b = ac \][/tex]
Divide both sides by [tex]\(a\)[/tex]:
[tex]\[ c = \frac{b}{a} \][/tex]
### 14. [tex]\(k = a - y; \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ k = a - y \][/tex]
Add [tex]\(y\)[/tex] to both sides:
[tex]\[ k + y = a \][/tex]
Subtract [tex]\(k\)[/tex] from both sides:
[tex]\[ y = a - k \][/tex]
### 15. [tex]\(dfg = h; \quad f\)[/tex]
To solve for [tex]\(f\)[/tex]:
[tex]\[ dfg = h \][/tex]
Divide both sides by [tex]\(dg\)[/tex]:
[tex]\[ f = \frac{h}{dg} \][/tex]
### 16. [tex]\(w = \frac{x}{a - b}; \quad x\)[/tex]
To solve for [tex]\(x\)[/tex]:
[tex]\[ w = \frac{x}{a - b} \][/tex]
Multiply both sides by [tex]\(a - b\)[/tex]:
[tex]\[ x = w(a - b) \][/tex]
### 17. [tex]\(2x + 3y = 12; \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ 2x + 3y = 12 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ 3y = 12 - 2x \][/tex]
Divide both sides by 3:
[tex]\[ y = \frac{12 - 2x}{3} \][/tex]
### 18. [tex]\(2n = 4x + 2y; \quad n\)[/tex]
To solve for [tex]\(n\)[/tex]:
[tex]\[ 2n = 4x + 2y \][/tex]
Divide both sides by 2:
[tex]\[ n = 2x + y \][/tex]
### 19. [tex]\(abc = \frac{1}{2}; \quad b\)[/tex]
To solve for [tex]\(b\)[/tex]:
[tex]\[ abc = \frac{1}{2} \][/tex]
Divide both sides by [tex]\(ac\)[/tex]:
[tex]\[ b = \frac{1}{2ac} \][/tex]
### 20. [tex]\(y = \frac{3}{5u} + 5; \quad u\)[/tex]
To solve for [tex]\(u\)[/tex]:
[tex]\[ y = \frac{3}{5u} + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ y - 5 = \frac{3}{5u} \][/tex]
Multiply both sides by [tex]\(5u\)[/tex]:
[tex]\[ 5u(y - 5) = 3 \][/tex]
Divide both sides by [tex]\(5(y - 5)\)[/tex]:
[tex]\[ u = \frac{3}{5(y - 5)} \][/tex]
### 21. [tex]\(8(x - a) = 2(2a - x); \quad x\)[/tex]
To solve for [tex]\(x\)[/tex]:
[tex]\[ 8(x - a) = 2(2a - x) \][/tex]
Expand both sides:
[tex]\[ 8x - 8a = 4a - 2x \][/tex]
Add [tex]\(2x\)[/tex] to both sides:
[tex]\[ 10x - 8a = 4a \][/tex]
Add [tex]\(8a\)[/tex] to both sides:
[tex]\[ 10x = 12a \][/tex]
Divide both sides by 10:
[tex]\[ x = \frac{6a}{5} \][/tex]
### 22. [tex]\(12(m + 3x) = 18(x - 3m); \quad m\)[/tex]
To solve for [tex]\(m\)[/tex]:
[tex]\[ 12(m + 3x) = 18(x - 3m) \][/tex]
Expand both sides:
[tex]\[ 12m + 36x = 18x - 54m \][/tex]
Add [tex]\(54m\)[/tex] to both sides:
[tex]\[ 66m + 36x = 18x \][/tex]
Subtract [tex]\(36x\)[/tex] from both sides:
[tex]\[ 66m = -18x \][/tex]
Divide both sides by 66:
[tex]\[ m = -\frac{18x}{66} = -\frac{x}{3.6667} = -\frac{x}{3.67} \][/tex]
### 23. [tex]\(V = \frac{1}{3}\pi r^2 h; \quad h\)[/tex]
To solve for [tex]\(h\)[/tex]:
[tex]\[ V = \frac{1}{3}\pi r^2 h \][/tex]
Multiply both sides by 3:
[tex]\[ 3V = \pi r^2 h \][/tex]
Divide both sides by [tex]\(\pi r^2\)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
### 24. [tex]\(V = \frac{1}{3}\pi r^2 (h - 1); \quad h\)[/tex]
To solve for [tex]\(h\)[/tex]:
[tex]\[ V = \frac{1}{3}\pi r^2 (h - 1) \][/tex]
Multiply both sides by 3:
[tex]\[ 3V = \pi r^2 (h - 1) \][/tex]
Divide both sides by [tex]\(\pi r^2\)[/tex]:
[tex]\[ \frac{3V}{\pi r^2} = h - 1 \][/tex]
Add 1 to both sides:
[tex]\[ h = \frac{3V}{\pi r^2} + 1 \][/tex]
### 25. [tex]\(y(a - b) = c(y + a); \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ y(a - b) = c(y + a) \][/tex]
Expand both sides:
[tex]\[ ya - yb = cy + ca \][/tex]
Move all [tex]\(y\)[/tex]-terms to one side:
[tex]\[ ya - yb - cy = ca \][/tex]
Factor out [tex]\(y\)[/tex]:
[tex]\[ y(a - b - c) = ca \][/tex]
Divide both sides by [tex]\((a - b - c)\)[/tex]:
[tex]\[ y = \frac{ca}{a - b - c} \][/tex]
### 26. [tex]\(x = \frac{3(y - b)}{m}; \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ x = \frac{3(y - b)}{m} \][/tex]
Multiply both sides by [tex]\(m\)[/tex]:
[tex]\[ xm = 3(y - b) \][/tex]
Divide both sides by 3:
[tex]\[ \frac{xm}{3} = y - b \][/tex]
Add [tex]\(b\)[/tex] to both sides:
[tex]\[ y = \frac{xm}{3} + b \][/tex]
### 27. [tex]\(F = -\frac{Gm}{r^2}; \quad G\)[/tex]
To solve for [tex]\(G\)[/tex]:
[tex]\[ F = -\frac{Gm}{r^2} \][/tex]
Multiply both sides by [tex]\(-r^2\)[/tex]:
[tex]\[ -Fr^2 = Gm \][/tex]
Divide both sides by [tex]\(m\)[/tex]:
[tex]\[ G = -\frac{Fr^2}{m} \][/tex]
These are the step-by-step solutions to the given equations.
### 13. [tex]\(\frac{b}{c} = a; \quad c\)[/tex]
To solve for [tex]\(c\)[/tex]:
[tex]\[ \frac{b}{c} = a \][/tex]
Multiply both sides by [tex]\(c\)[/tex]:
[tex]\[ b = ac \][/tex]
Divide both sides by [tex]\(a\)[/tex]:
[tex]\[ c = \frac{b}{a} \][/tex]
### 14. [tex]\(k = a - y; \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ k = a - y \][/tex]
Add [tex]\(y\)[/tex] to both sides:
[tex]\[ k + y = a \][/tex]
Subtract [tex]\(k\)[/tex] from both sides:
[tex]\[ y = a - k \][/tex]
### 15. [tex]\(dfg = h; \quad f\)[/tex]
To solve for [tex]\(f\)[/tex]:
[tex]\[ dfg = h \][/tex]
Divide both sides by [tex]\(dg\)[/tex]:
[tex]\[ f = \frac{h}{dg} \][/tex]
### 16. [tex]\(w = \frac{x}{a - b}; \quad x\)[/tex]
To solve for [tex]\(x\)[/tex]:
[tex]\[ w = \frac{x}{a - b} \][/tex]
Multiply both sides by [tex]\(a - b\)[/tex]:
[tex]\[ x = w(a - b) \][/tex]
### 17. [tex]\(2x + 3y = 12; \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ 2x + 3y = 12 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ 3y = 12 - 2x \][/tex]
Divide both sides by 3:
[tex]\[ y = \frac{12 - 2x}{3} \][/tex]
### 18. [tex]\(2n = 4x + 2y; \quad n\)[/tex]
To solve for [tex]\(n\)[/tex]:
[tex]\[ 2n = 4x + 2y \][/tex]
Divide both sides by 2:
[tex]\[ n = 2x + y \][/tex]
### 19. [tex]\(abc = \frac{1}{2}; \quad b\)[/tex]
To solve for [tex]\(b\)[/tex]:
[tex]\[ abc = \frac{1}{2} \][/tex]
Divide both sides by [tex]\(ac\)[/tex]:
[tex]\[ b = \frac{1}{2ac} \][/tex]
### 20. [tex]\(y = \frac{3}{5u} + 5; \quad u\)[/tex]
To solve for [tex]\(u\)[/tex]:
[tex]\[ y = \frac{3}{5u} + 5 \][/tex]
Subtract 5 from both sides:
[tex]\[ y - 5 = \frac{3}{5u} \][/tex]
Multiply both sides by [tex]\(5u\)[/tex]:
[tex]\[ 5u(y - 5) = 3 \][/tex]
Divide both sides by [tex]\(5(y - 5)\)[/tex]:
[tex]\[ u = \frac{3}{5(y - 5)} \][/tex]
### 21. [tex]\(8(x - a) = 2(2a - x); \quad x\)[/tex]
To solve for [tex]\(x\)[/tex]:
[tex]\[ 8(x - a) = 2(2a - x) \][/tex]
Expand both sides:
[tex]\[ 8x - 8a = 4a - 2x \][/tex]
Add [tex]\(2x\)[/tex] to both sides:
[tex]\[ 10x - 8a = 4a \][/tex]
Add [tex]\(8a\)[/tex] to both sides:
[tex]\[ 10x = 12a \][/tex]
Divide both sides by 10:
[tex]\[ x = \frac{6a}{5} \][/tex]
### 22. [tex]\(12(m + 3x) = 18(x - 3m); \quad m\)[/tex]
To solve for [tex]\(m\)[/tex]:
[tex]\[ 12(m + 3x) = 18(x - 3m) \][/tex]
Expand both sides:
[tex]\[ 12m + 36x = 18x - 54m \][/tex]
Add [tex]\(54m\)[/tex] to both sides:
[tex]\[ 66m + 36x = 18x \][/tex]
Subtract [tex]\(36x\)[/tex] from both sides:
[tex]\[ 66m = -18x \][/tex]
Divide both sides by 66:
[tex]\[ m = -\frac{18x}{66} = -\frac{x}{3.6667} = -\frac{x}{3.67} \][/tex]
### 23. [tex]\(V = \frac{1}{3}\pi r^2 h; \quad h\)[/tex]
To solve for [tex]\(h\)[/tex]:
[tex]\[ V = \frac{1}{3}\pi r^2 h \][/tex]
Multiply both sides by 3:
[tex]\[ 3V = \pi r^2 h \][/tex]
Divide both sides by [tex]\(\pi r^2\)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
### 24. [tex]\(V = \frac{1}{3}\pi r^2 (h - 1); \quad h\)[/tex]
To solve for [tex]\(h\)[/tex]:
[tex]\[ V = \frac{1}{3}\pi r^2 (h - 1) \][/tex]
Multiply both sides by 3:
[tex]\[ 3V = \pi r^2 (h - 1) \][/tex]
Divide both sides by [tex]\(\pi r^2\)[/tex]:
[tex]\[ \frac{3V}{\pi r^2} = h - 1 \][/tex]
Add 1 to both sides:
[tex]\[ h = \frac{3V}{\pi r^2} + 1 \][/tex]
### 25. [tex]\(y(a - b) = c(y + a); \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ y(a - b) = c(y + a) \][/tex]
Expand both sides:
[tex]\[ ya - yb = cy + ca \][/tex]
Move all [tex]\(y\)[/tex]-terms to one side:
[tex]\[ ya - yb - cy = ca \][/tex]
Factor out [tex]\(y\)[/tex]:
[tex]\[ y(a - b - c) = ca \][/tex]
Divide both sides by [tex]\((a - b - c)\)[/tex]:
[tex]\[ y = \frac{ca}{a - b - c} \][/tex]
### 26. [tex]\(x = \frac{3(y - b)}{m}; \quad y\)[/tex]
To solve for [tex]\(y\)[/tex]:
[tex]\[ x = \frac{3(y - b)}{m} \][/tex]
Multiply both sides by [tex]\(m\)[/tex]:
[tex]\[ xm = 3(y - b) \][/tex]
Divide both sides by 3:
[tex]\[ \frac{xm}{3} = y - b \][/tex]
Add [tex]\(b\)[/tex] to both sides:
[tex]\[ y = \frac{xm}{3} + b \][/tex]
### 27. [tex]\(F = -\frac{Gm}{r^2}; \quad G\)[/tex]
To solve for [tex]\(G\)[/tex]:
[tex]\[ F = -\frac{Gm}{r^2} \][/tex]
Multiply both sides by [tex]\(-r^2\)[/tex]:
[tex]\[ -Fr^2 = Gm \][/tex]
Divide both sides by [tex]\(m\)[/tex]:
[tex]\[ G = -\frac{Fr^2}{m} \][/tex]
These are the step-by-step solutions to the given equations.
