Sure, I will provide a step-by-step solution for each of the three equations given.
### Equation 1: [tex]\(\quad \frac{2m + 5}{3} = 3m - 18\)[/tex]
1. Clear the fraction by multiplying both sides of the equation by 3:
[tex]\[
2m + 5 = 9m - 54
\][/tex]
2. Rearrange the equation to bring all terms containing [tex]\( m \)[/tex] to one side and constants to the other:
[tex]\[
2m - 9m = -54 - 5
\][/tex]
3. Combine like terms:
[tex]\[
-7m = -59
\][/tex]
4. Solve for [tex]\( m \)[/tex] by dividing both sides by -7:
[tex]\[
m = \frac{59}{7}
\][/tex]
### Equation 2: [tex]\(\quad \frac{x}{2} + \frac{x}{4} = \frac{1}{8}\)[/tex]
1. Find a common denominator for the fractions on the left side:
[tex]\[
\frac{2x}{4} + \frac{x}{4} = \frac{1}{8}
\][/tex]
2. Combine like terms within the fractions:
[tex]\[
\frac{3x}{4} = \frac{1}{8}
\][/tex]
3. Clear the fraction by multiplying both sides of the equation by 8:
[tex]\[
8 \cdot \frac{3x}{4} = 8 \cdot \frac{1}{8}
\][/tex]
4. Simplify both sides:
[tex]\[
6x = 1
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 6:
[tex]\[
x = \frac{1}{6}
\][/tex]
### Equation 3: [tex]\(\quad 2z + \frac{8}{3} = \frac{1}{4}z + 5\)[/tex]
1. Clear the constants by combining like terms:
[tex]\[
2z - \frac{1}{4}z = 5 - \frac{8}{3}
\][/tex]
2. Find a common denominator for the fractions:
[tex]\[
\frac{8z}{4} - \frac{1z}{4} = 5 - \frac{8}{3}
\][/tex]
3. Combine like terms on both sides:
[tex]\[
\frac{7z}{4} = 5 - \frac{8}{3}
\][/tex]
4. Simplify the right-hand side by finding a common denominator:
[tex]\[
5 - \frac{8}{3} = \frac{15}{3} - \frac{8}{3} = \frac{7}{3}
\][/tex]
5. Equate both simplified sides:
[tex]\[
\frac{7z}{4} = \frac{7}{3}
\][/tex]
6. Cross-multiply to clear the fractions:
[tex]\[
7z \cdot 3 = 7 \cdot 4
\][/tex]
7. Simplify the resulting equation:
[tex]\[
21z = 28
\][/tex]
8. Solve for [tex]\( z \)[/tex] by dividing both sides by 21:
[tex]\[
z = \frac{28}{21} = \frac{4}{3}
\][/tex]
Thus, the solutions to the equations are:
[tex]\[
m = \frac{59}{7}, \quad x = \frac{1}{6}, \quad z = \frac{4}{3}
\][/tex]
Or in decimal form:
[tex]\[
m \approx 8.42857, \quad x \approx 0.16667, \quad z \approx 1.33333
\][/tex]
