Answer :
Sure, let's solve these equations one by one, step-by-step:
### a) [tex]\(\frac{x}{10} = \frac{3}{6} - 5\)[/tex]:
1. Simplify [tex]\(\frac{3}{6}\)[/tex] to [tex]\(0.5\)[/tex]:
[tex]\[
\frac{x}{10} = 0.5 - 5
\][/tex]
2. Subtract 5 from 0.5:
[tex]\[
\frac{x}{10} = -4.5
\][/tex]
3. Multiply both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4.5 \times 10 = -45
\][/tex]
Solution for a): [tex]\( x = -45 \)[/tex]
### b) [tex]\(\frac{5}{x} = \frac{15}{6}\)[/tex]:
1. Simplify [tex]\(\frac{15}{6}\)[/tex] to [tex]\(2.5\)[/tex]:
[tex]\[
\frac{5}{x} = 2.5
\][/tex]
2. To solve for [tex]\(x\)[/tex], multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
5 = 2.5x
\][/tex]
3. Divide both sides by 2.5:
[tex]\[
x = \frac{5}{2.5} = 2
\][/tex]
Solution for b): [tex]\( x = 2 \)[/tex]
### c) [tex]\(\frac{3}{x} = \frac{x}{48}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
3 \times 48 = x \times x
\][/tex]
2. Simplify:
[tex]\[
144 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 12
\][/tex]
Solution for c): [tex]\( x = -12, 12 \)[/tex]
### d) [tex]\(\frac{x}{9} = \frac{9}{27}\)[/tex]:
1. Simplify [tex]\(\frac{9}{27}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{x}{9} = \frac{1}{3}
\][/tex]
2. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 9 \times \frac{1}{3} = 3
\][/tex]
Solution for d): [tex]\( x = 3 \)[/tex]
### e) [tex]\(\frac{25}{x} = \frac{x}{4}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
25 \times 4 = x \times x
\][/tex]
2. Simplify:
[tex]\[
100 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 10
\][/tex]
Solution for e): [tex]\( x = -10, 10 \)[/tex]
### f) [tex]\(\frac{48}{12} = \frac{12}{x}\)[/tex]:
1. Simplify [tex]\(\frac{48}{12}\)[/tex] to 4:
[tex]\[
4 = \frac{12}{x}
\][/tex]
2. Multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
4x = 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
x = \frac{12}{4} = 3
\][/tex]
Solution for f): [tex]\( x = 3 \)[/tex]
### g) [tex]\(\frac{3}{x} = \frac{14}{6}\)[/tex]:
1. Simplify [tex]\(\frac{14}{6}\)[/tex] to [tex]\(\frac{7}{3}\)[/tex]:
[tex]\[
\frac{3}{x} = \frac{7}{3}
\][/tex]
2. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
3 \times 3 = 7x
\][/tex]
3. Simplify:
[tex]\[
9 = 7x
\][/tex]
4. Divide both sides by 7:
[tex]\[
x = \frac{9}{7} \approx 1.28571428571429
\][/tex]
Solution for g): [tex]\( x \approx 1.28571428571429 \)[/tex]
### h) [tex]\(\frac{6}{5} = \frac{x}{10}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
6 \times 10 = 5x
\][/tex]
2. Simplify:
[tex]\[
60 = 5x
\][/tex]
3. Divide both sides by 5:
[tex]\[
x = \frac{60}{5} = 12
\][/tex]
Solution for h): [tex]\( x = 12 \)[/tex]
### i) [tex]\(\frac{4}{x} = \frac{x}{16}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
4 \times 16 = x^2
\][/tex]
2. Simplify:
[tex]\[
64 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 8
\][/tex]
Solution for i): [tex]\( x = -8, 8 \)[/tex]
### j) [tex]\(\frac{25}{5} = \frac{x}{2}\)[/tex]:
1. Simplify [tex]\(\frac{25}{5}\)[/tex] to 5:
[tex]\[
5 = \frac{x}{2}
\][/tex]
2. Multiply both sides by 2:
[tex]\[
x = 5 \times 2 = 10
\][/tex]
Solution for j): [tex]\( x = 10 \)[/tex]
### k) [tex]\(\frac{8}{4} = \frac{6}{x}\)[/tex]:
1. Simplify [tex]\(\frac{8}{4}\)[/tex] to 2:
[tex]\[
2 = \frac{6}{x}
\][/tex]
2. Multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
2x = 6
\][/tex]
3. Divide both sides by 2:
[tex]\[
x = \frac{6}{2} = 3
\][/tex]
Solution for k): [tex]\( x = 3 \)[/tex]
### l) [tex]\(\frac{32}{x} = \frac{x}{50}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
32 \times 50 = x^2
\][/tex]
2. Simplify:
[tex]\[
1600 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 40
\][/tex]
Solution for l): [tex]\( x = -40, 40 \)[/tex]
These are the solutions for each equation:
[tex]\[
\begin{align*}
\text{a)} & \quad x = -45 \\
\text{b)} & \quad x = 2 \\
\text{c)} & \quad x = -12, 12 \\
\text{d)} & \quad x = 3 \\
\text{e)} & \quad x = -10, 10 \\
\text{f)} & \quad x = 3 \\
\text{g)} & \quad x \approx 1.28571428571429 \\
\text{h)} & \quad x = 12 \\
\text{i)} & \quad x = -8, 8 \\
\text{j)} & \quad x = 10 \\
\text{k)} & \quad x = 3 \\
\text{l)} & \quad x = -40, 40 \\
\end{align*}
\][/tex]
### a) [tex]\(\frac{x}{10} = \frac{3}{6} - 5\)[/tex]:
1. Simplify [tex]\(\frac{3}{6}\)[/tex] to [tex]\(0.5\)[/tex]:
[tex]\[
\frac{x}{10} = 0.5 - 5
\][/tex]
2. Subtract 5 from 0.5:
[tex]\[
\frac{x}{10} = -4.5
\][/tex]
3. Multiply both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4.5 \times 10 = -45
\][/tex]
Solution for a): [tex]\( x = -45 \)[/tex]
### b) [tex]\(\frac{5}{x} = \frac{15}{6}\)[/tex]:
1. Simplify [tex]\(\frac{15}{6}\)[/tex] to [tex]\(2.5\)[/tex]:
[tex]\[
\frac{5}{x} = 2.5
\][/tex]
2. To solve for [tex]\(x\)[/tex], multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
5 = 2.5x
\][/tex]
3. Divide both sides by 2.5:
[tex]\[
x = \frac{5}{2.5} = 2
\][/tex]
Solution for b): [tex]\( x = 2 \)[/tex]
### c) [tex]\(\frac{3}{x} = \frac{x}{48}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
3 \times 48 = x \times x
\][/tex]
2. Simplify:
[tex]\[
144 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 12
\][/tex]
Solution for c): [tex]\( x = -12, 12 \)[/tex]
### d) [tex]\(\frac{x}{9} = \frac{9}{27}\)[/tex]:
1. Simplify [tex]\(\frac{9}{27}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{x}{9} = \frac{1}{3}
\][/tex]
2. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 9 \times \frac{1}{3} = 3
\][/tex]
Solution for d): [tex]\( x = 3 \)[/tex]
### e) [tex]\(\frac{25}{x} = \frac{x}{4}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
25 \times 4 = x \times x
\][/tex]
2. Simplify:
[tex]\[
100 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 10
\][/tex]
Solution for e): [tex]\( x = -10, 10 \)[/tex]
### f) [tex]\(\frac{48}{12} = \frac{12}{x}\)[/tex]:
1. Simplify [tex]\(\frac{48}{12}\)[/tex] to 4:
[tex]\[
4 = \frac{12}{x}
\][/tex]
2. Multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
4x = 12
\][/tex]
3. Divide both sides by 4:
[tex]\[
x = \frac{12}{4} = 3
\][/tex]
Solution for f): [tex]\( x = 3 \)[/tex]
### g) [tex]\(\frac{3}{x} = \frac{14}{6}\)[/tex]:
1. Simplify [tex]\(\frac{14}{6}\)[/tex] to [tex]\(\frac{7}{3}\)[/tex]:
[tex]\[
\frac{3}{x} = \frac{7}{3}
\][/tex]
2. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
3 \times 3 = 7x
\][/tex]
3. Simplify:
[tex]\[
9 = 7x
\][/tex]
4. Divide both sides by 7:
[tex]\[
x = \frac{9}{7} \approx 1.28571428571429
\][/tex]
Solution for g): [tex]\( x \approx 1.28571428571429 \)[/tex]
### h) [tex]\(\frac{6}{5} = \frac{x}{10}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
6 \times 10 = 5x
\][/tex]
2. Simplify:
[tex]\[
60 = 5x
\][/tex]
3. Divide both sides by 5:
[tex]\[
x = \frac{60}{5} = 12
\][/tex]
Solution for h): [tex]\( x = 12 \)[/tex]
### i) [tex]\(\frac{4}{x} = \frac{x}{16}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
4 \times 16 = x^2
\][/tex]
2. Simplify:
[tex]\[
64 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 8
\][/tex]
Solution for i): [tex]\( x = -8, 8 \)[/tex]
### j) [tex]\(\frac{25}{5} = \frac{x}{2}\)[/tex]:
1. Simplify [tex]\(\frac{25}{5}\)[/tex] to 5:
[tex]\[
5 = \frac{x}{2}
\][/tex]
2. Multiply both sides by 2:
[tex]\[
x = 5 \times 2 = 10
\][/tex]
Solution for j): [tex]\( x = 10 \)[/tex]
### k) [tex]\(\frac{8}{4} = \frac{6}{x}\)[/tex]:
1. Simplify [tex]\(\frac{8}{4}\)[/tex] to 2:
[tex]\[
2 = \frac{6}{x}
\][/tex]
2. Multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
2x = 6
\][/tex]
3. Divide both sides by 2:
[tex]\[
x = \frac{6}{2} = 3
\][/tex]
Solution for k): [tex]\( x = 3 \)[/tex]
### l) [tex]\(\frac{32}{x} = \frac{x}{50}\)[/tex]:
1. Cross multiply to solve for [tex]\(x\)[/tex]:
[tex]\[
32 \times 50 = x^2
\][/tex]
2. Simplify:
[tex]\[
1600 = x^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
x = \pm 40
\][/tex]
Solution for l): [tex]\( x = -40, 40 \)[/tex]
These are the solutions for each equation:
[tex]\[
\begin{align*}
\text{a)} & \quad x = -45 \\
\text{b)} & \quad x = 2 \\
\text{c)} & \quad x = -12, 12 \\
\text{d)} & \quad x = 3 \\
\text{e)} & \quad x = -10, 10 \\
\text{f)} & \quad x = 3 \\
\text{g)} & \quad x \approx 1.28571428571429 \\
\text{h)} & \quad x = 12 \\
\text{i)} & \quad x = -8, 8 \\
\text{j)} & \quad x = 10 \\
\text{k)} & \quad x = 3 \\
\text{l)} & \quad x = -40, 40 \\
\end{align*}
\][/tex]
