Answer :
To determine which equation gives the value of [tex]\( x = 8 \)[/tex] when solved, we will go through each option step-by-step, solving for [tex]\( x \)[/tex].
### Option A: [tex]\(\frac{5}{2} x + \frac{7}{2} = \frac{3}{4} x + 14\)[/tex]
1. Subtract [tex]\(\frac{3}{4} x\)[/tex] from both sides:
[tex]\[ \frac{5}{2} x - \frac{3}{4} x + \frac{7}{2} = 14 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \left( \frac{5}{2} - \frac{3}{4} \right) x + \frac{7}{2} = 14 \][/tex]
[tex]\[ \frac{10}{4} x - \frac{3}{4} x + \frac{7}{2} = 14 \][/tex]
[tex]\[ \frac{7}{4} x + \frac{7}{2} = 14 \][/tex]
3. Subtract [tex]\(\frac{7}{2}\)[/tex] from both sides:
[tex]\[ \frac{7}{4} x = 14 - \frac{7}{2} \][/tex]
[tex]\[ \frac{7}{4} x = \frac{28}{2} - \frac{7}{2} \][/tex]
[tex]\[ \frac{7}{4} x = \frac{21}{2} \][/tex]
4. Multiply both sides by [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ x = \frac{21}{2} \cdot \frac{4}{7} \][/tex]
[tex]\[ x = 3 \times 2 \][/tex]
[tex]\[ x = 6 \][/tex]
Option A does not give [tex]\( x = 8 \)[/tex].
### Option B: [tex]\(\frac{5}{4} x - 9 = \frac{3}{2} x - 12\)[/tex]
1. Subtract [tex]\(\frac{3}{2} x\)[/tex] from both sides:
[tex]\[ \frac{5}{4} x - \frac{3}{2} x - 9 = -12 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{5}{4} x - \frac{6}{4} x - 9 = -12 \][/tex]
[tex]\[ -\frac{1}{4} x - 9 = -12 \][/tex]
3. Add 9 to both sides:
[tex]\[ -\frac{1}{4} x = -12 + 9 \][/tex]
[tex]\[ -\frac{1}{4} x = -3 \][/tex]
4. Multiply both sides by [tex]\(-4\)[/tex]:
[tex]\[ x = -3 \times -4 \][/tex]
[tex]\[ x = 12 \][/tex]
Option B does not give [tex]\( x = 8 \)[/tex].
### Option C: [tex]\(\frac{5}{4} x - 2 = \frac{3}{2} x - 4\)[/tex]
1. Subtract [tex]\(\frac{3}{2} x\)[/tex] from both sides:
[tex]\[ \frac{5}{4} x - \frac{3}{2} x - 2 = -4 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{5}{4} x - \frac{6}{4} x - 2 = -4 \][/tex]
[tex]\[ -\frac{1}{4} x - 2 = -4 \][/tex]
3. Add 2 to both sides:
[tex]\[ -\frac{1}{4} x = -4 + 2 \][/tex]
[tex]\[ -\frac{1}{4} x = -2 \][/tex]
4. Multiply both sides by [tex]\(-4\)[/tex]:
[tex]\[ x = -2 \times -4 \][/tex]
[tex]\[ x = 8 \][/tex]
Option C gives [tex]\( x = 8 \)[/tex].
### Option D: [tex]\(\frac{5}{2} x - 7 = \frac{3}{4} x + 14\)[/tex]
1. Subtract [tex]\(\frac{3}{4} x\)[/tex] from both sides:
[tex]\[ \frac{5}{2} x - \frac{3}{4} x - 7 = 14 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{10}{4} x - \frac{3}{4} x - 7 = 14 \][/tex]
[tex]\[ \frac{7}{4} x - 7 = 14 \][/tex]
3. Add 7 to both sides:
[tex]\[ \frac{7}{4} x = 14 + 7 \][/tex]
[tex]\[ \frac{7}{4} x = 21 \][/tex]
4. Multiply both sides by [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ x = 21 \times \frac{4}{7} \][/tex]
[tex]\[ x = 3 \times 4 \][/tex]
[tex]\[ x = 12 \][/tex]
Option D does not give [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
### Option A: [tex]\(\frac{5}{2} x + \frac{7}{2} = \frac{3}{4} x + 14\)[/tex]
1. Subtract [tex]\(\frac{3}{4} x\)[/tex] from both sides:
[tex]\[ \frac{5}{2} x - \frac{3}{4} x + \frac{7}{2} = 14 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \left( \frac{5}{2} - \frac{3}{4} \right) x + \frac{7}{2} = 14 \][/tex]
[tex]\[ \frac{10}{4} x - \frac{3}{4} x + \frac{7}{2} = 14 \][/tex]
[tex]\[ \frac{7}{4} x + \frac{7}{2} = 14 \][/tex]
3. Subtract [tex]\(\frac{7}{2}\)[/tex] from both sides:
[tex]\[ \frac{7}{4} x = 14 - \frac{7}{2} \][/tex]
[tex]\[ \frac{7}{4} x = \frac{28}{2} - \frac{7}{2} \][/tex]
[tex]\[ \frac{7}{4} x = \frac{21}{2} \][/tex]
4. Multiply both sides by [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ x = \frac{21}{2} \cdot \frac{4}{7} \][/tex]
[tex]\[ x = 3 \times 2 \][/tex]
[tex]\[ x = 6 \][/tex]
Option A does not give [tex]\( x = 8 \)[/tex].
### Option B: [tex]\(\frac{5}{4} x - 9 = \frac{3}{2} x - 12\)[/tex]
1. Subtract [tex]\(\frac{3}{2} x\)[/tex] from both sides:
[tex]\[ \frac{5}{4} x - \frac{3}{2} x - 9 = -12 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{5}{4} x - \frac{6}{4} x - 9 = -12 \][/tex]
[tex]\[ -\frac{1}{4} x - 9 = -12 \][/tex]
3. Add 9 to both sides:
[tex]\[ -\frac{1}{4} x = -12 + 9 \][/tex]
[tex]\[ -\frac{1}{4} x = -3 \][/tex]
4. Multiply both sides by [tex]\(-4\)[/tex]:
[tex]\[ x = -3 \times -4 \][/tex]
[tex]\[ x = 12 \][/tex]
Option B does not give [tex]\( x = 8 \)[/tex].
### Option C: [tex]\(\frac{5}{4} x - 2 = \frac{3}{2} x - 4\)[/tex]
1. Subtract [tex]\(\frac{3}{2} x\)[/tex] from both sides:
[tex]\[ \frac{5}{4} x - \frac{3}{2} x - 2 = -4 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{5}{4} x - \frac{6}{4} x - 2 = -4 \][/tex]
[tex]\[ -\frac{1}{4} x - 2 = -4 \][/tex]
3. Add 2 to both sides:
[tex]\[ -\frac{1}{4} x = -4 + 2 \][/tex]
[tex]\[ -\frac{1}{4} x = -2 \][/tex]
4. Multiply both sides by [tex]\(-4\)[/tex]:
[tex]\[ x = -2 \times -4 \][/tex]
[tex]\[ x = 8 \][/tex]
Option C gives [tex]\( x = 8 \)[/tex].
### Option D: [tex]\(\frac{5}{2} x - 7 = \frac{3}{4} x + 14\)[/tex]
1. Subtract [tex]\(\frac{3}{4} x\)[/tex] from both sides:
[tex]\[ \frac{5}{2} x - \frac{3}{4} x - 7 = 14 \][/tex]
2. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{10}{4} x - \frac{3}{4} x - 7 = 14 \][/tex]
[tex]\[ \frac{7}{4} x - 7 = 14 \][/tex]
3. Add 7 to both sides:
[tex]\[ \frac{7}{4} x = 14 + 7 \][/tex]
[tex]\[ \frac{7}{4} x = 21 \][/tex]
4. Multiply both sides by [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ x = 21 \times \frac{4}{7} \][/tex]
[tex]\[ x = 3 \times 4 \][/tex]
[tex]\[ x = 12 \][/tex]
Option D does not give [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
