Let's solve each of these equations step by step to find the values of [tex]\(x\)[/tex] and [tex]\(a\)[/tex].
### Solve Equation 1:
[tex]\[ 2 \cdot \frac{4x - 4}{8} - \frac{4x - 2}{6} = \frac{1}{2} \][/tex]
1. Simplify the terms:
[tex]\[
2 \cdot \frac{4x - 4}{8} = \frac{4x - 4}{4} = x - 1
\][/tex]
[tex]\[
- \frac{4x - 2}{6} = - \frac{2(2x - 1)}{6} = - \frac{2x - 1}{3}
\][/tex]
2. Combine the simplified terms:
[tex]\[
x - 1 - \frac{2x - 1}{3} = \frac{1}{2}
\][/tex]
3. Find a common denominator and combine:
[tex]\[
\left( \frac{3(x - 1) - (2x - 1)}{3} \right) = \frac{1}{2}
\][/tex]
[tex]\[
\left( \frac{3x - 3 - 2x + 1}{3} \right) = \frac{1}{2}
\][/tex]
[tex]\[
\frac{x - 2}{3} = \frac{1}{2}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
2(x - 2) = 3
\][/tex]
[tex]\[
2x - 4 = 3
\][/tex]
[tex]\[
2x = 7
\][/tex]
[tex]\[
x = \frac{7}{2} = 3.5
\][/tex]
So, the solution for the first equation is:
[tex]\[ x = 3.5 \][/tex]
### Solve Equation 2:
[tex]\[ \frac{3x + 1}{3} = \frac{x + 4}{2} \][/tex]
1. Clear the fractions by finding a common denominator:
[tex]\[
2(3x + 1) = 3(x + 4)
\][/tex]
[tex]\[
6x + 2 = 3x + 12
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
6x - 3x = 12 - 2
\][/tex]
[tex]\[
3x = 10
\][/tex]
[tex]\[
x = \frac{10}{3}
\][/tex]
So, the solution for the second equation is:
[tex]\[ x = \frac{10}{3} \][/tex]
### Solve Equation 3:
[tex]\[ 4 \frac{2x - 2}{8} - \frac{2x - 4}{8} = 9 \][/tex]
1. Simplify the terms:
[tex]\[
4 \cdot \frac{2x - 2}{8} = \frac{4(2x - 2)}{8} = \frac{8x - 8}{8} = x - 1
\][/tex]
[tex]\[
- \frac{2x - 4}{8} = - \frac{2x - 4}{8}
\][/tex]
2. Combine the terms:
[tex]\[
x - 1 - \frac{2x - 4}{8} = 9
\][/tex]
3. Clear the fraction:
[tex]\[
x - 1 - \frac{2x - 4}{8} = 9
\][/tex]
[tex]\[
8(x - 1) - (2x - 4) = 72
\][/tex]
[tex]\[
8x - 8 - 2x + 4 = 72
\][/tex]
[tex]\[
6x - 4 = 72
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
6x = 76
\][/tex]
[tex]\[
x = \frac{76}{6} = \frac{38}{3}
\][/tex]
So, the solution for the third equation is:
[tex]\[ x = \frac{38}{3} \][/tex]
### Solve Equation 4:
[tex]\[ \frac{3x}{2} \cdot \frac{-15}{2} = \frac{2a}{3} - \frac{5}{3} \][/tex]
1. Simplify the left-hand side:
[tex]\[
\frac{3x}{2} \cdot \frac{-15}{2} = \frac{3x \cdot -15}{4} = -\frac{45x}{4}
\][/tex]
2. Write the equation:
[tex]\[
-\frac{45x}{4} = \frac{2a}{3} - \frac{5}{3}
\][/tex]
3. Clear the fractions by finding a common denominator:
[tex]\[
-\frac{45x}{4} = \frac{2a - 5}{3}
\][/tex]
4. Isolate [tex]\(a\)[/tex]:
[tex]\[
3 \cdot \left(-\frac{45x}{4}\right) = 2a - 5
\][/tex]
[tex]\[
-\frac{135x}{4} = 2a - 5
\][/tex]
[tex]\[
2a = -\frac{135x}{4} + 5
\][/tex]
[tex]\[
a = -\frac{135x}{8} + \frac{5}{2}
\][/tex]
So, the solution for the fourth equation in terms of [tex]\(a\)[/tex] is:
[tex]\[ a = -\frac{135x}{8} + \frac{5}{2} \][/tex]
Or more concisely:
[tex]\[ a = 2.5 - 16.875x \][/tex]
### Final Solutions:
The solutions for the equations are:
1. [tex]\( x = 3.5 \)[/tex]
2. [tex]\( x = \frac{10}{3} \)[/tex]
3. [tex]\( x = \frac{38}{3} \)[/tex]
4. [tex]\( a = 2.5 - 16.875x \)[/tex]
