Sure, let's solve each of the equations one by one.
### Equation (i)
The first equation to solve is:
[tex]\[ -25x + 74 = 6(3x - 2) \][/tex]
1. Distribute the 6 on the right-hand side:
[tex]\[ -25x + 74 = 18x - 12 \][/tex]
2. Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -25x - 18x = -12 - 74 \][/tex]
[tex]\[ -43x = -86 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-86}{-43} \][/tex]
[tex]\[ x = 2 \][/tex]
So, the solution for equation (i) is:
[tex]\[ x = 2 \][/tex]
### Equation (ii)
The second equation is:
[tex]\[ \frac{2x}{3x - 2} = \frac{-3}{5} \][/tex]
1. Cross-multiply to get rid of the fraction:
[tex]\[ 2x \cdot 5 = -3 \cdot (3x - 2) \][/tex]
[tex]\[ 10x = -9x + 6 \][/tex]
2. Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 10x + 9x = 6 \][/tex]
[tex]\[ 19x = 6 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{6}{19} \][/tex]
[tex]\[ x \approx 0.315789 \][/tex]
So, the solution for equation (ii) is:
[tex]\[ x \approx 0.315789 \][/tex]
### Equation (iii)
The third equation is:
[tex]\[ \frac{3x^2 - 4x}{6} = 20x \][/tex]
1. Multiply both sides by 6 to get rid of the fraction:
[tex]\[ 3x^2 - 4x = 120x \][/tex]
2. Move all terms to one side to set the equation to zero:
[tex]\[ 3x^2 - 4x - 120x = 0 \][/tex]
[tex]\[ 3x^2 - 124x = 0 \][/tex]
3. Factor out the common term [tex]\(x\)[/tex]:
[tex]\[ x(3x - 124) = 0 \][/tex]
4. Set each factor equal to zero:
[tex]\[ x = 0 \][/tex]
[tex]\[ 3x - 124 = 0 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[ 3x = 124 \][/tex]
[tex]\[ x = \frac{124}{3} \][/tex]
[tex]\[ x \approx 41.3333 \][/tex]
So, the solutions for equation (iii) are:
[tex]\[ x = 0 \text{ or } x = \frac{124}{3} \][/tex]
### Summary of Solutions
- For equation (i): [tex]\( x = 2 \)[/tex]
- For equation (ii): [tex]\( x \approx 0.315789 \)[/tex]
- For equation (iii): [tex]\( x = 0 \text{ or } x = \frac{124}{3} \)[/tex]
