To determine which of the provided equations results in [tex]\( x = 8 \)[/tex], we'll solve each of them step-by-step and verify the solution.
### Equation A
[tex]\[
\frac{5}{2} x + \frac{7}{2} = \frac{3}{4} x + 14
\][/tex]
Step 1: Substitute [tex]\( x = 8 \)[/tex] into the equation.
[tex]\[
\frac{5}{2} \cdot 8 + \frac{7}{2} = \frac{3}{4} \cdot 8 + 14
\][/tex]
Step 2: Calculate the left-hand side (LHS) and right-hand side (RHS) separately.
LHS:
[tex]\[
\frac{5}{2} \cdot 8 = 20, \quad \text{and} \quad \frac{7}{2} = 3.5
\][/tex]
[tex]\[
20 + 3.5 = 23.5
\][/tex]
RHS:
[tex]\[
\frac{3}{4} \cdot 8 = 6, \quad \text{so} \quad 6 + 14 = 20
\][/tex]
Clearly, [tex]\( 23.5 \neq 20 \)[/tex].
### Equation B
[tex]\[
\frac{5}{4} x - 9 = \frac{3}{2} x - 12
\][/tex]
Step 1: Substitute [tex]\( x = 8 \)[/tex] into the equation.
[tex]\[
\frac{5}{4} \cdot 8 - 9 = \frac{3}{2} \cdot 8 - 12
\][/tex]
Step 2: Calculate the LHS and RHS.
LHS:
[tex]\[
\frac{5}{4} \cdot 8 = 10, \quad \text{so} \quad 10 - 9 = 1
\][/tex]
RHS:
[tex]\[
\frac{3}{2} \cdot 8 = 12, \quad \text{so} \quad 12 - 12 = 0
\][/tex]
Clearly, [tex]\( 1 \neq 0 \)[/tex].
### Equation C
[tex]\[
\frac{5}{4} x - 2 = \frac{3}{2} x - 4
\][/tex]
Step 1: Substitute [tex]\( x = 8 \)[/tex] into the equation.
[tex]\[
\frac{5}{4} \cdot 8 - 2 = \frac{3}{2} \cdot 8 - 4
\][/tex]
Step 2: Calculate the LHS and RHS.
LHS:
[tex]\[
\frac{5}{4} \cdot 8 = 10, \quad \text{so} \quad 10 - 2 = 8
\][/tex]
RHS:
[tex]\[
\frac{3}{2} \cdot 8 = 12, \quad \text{so} \quad 12 - 4 = 8
\][/tex]
Clearly, [tex]\( 8 = 8 \)[/tex].
### Equation D
[tex]\[
\frac{5}{2} x - 7 = \frac{3}{4} x + 14
\][/tex]
Step 1: Substitute [tex]\( x = 8 \)[/tex] into the equation.
[tex]\[
\frac{5}{2} \cdot 8 - 7 = \frac{3}{4} \cdot 8 + 14
\][/tex]
Step 2: Calculate the LHS and RHS.
LHS:
[tex]\[
\frac{5}{2} \cdot 8 = 20, \quad \text{so} \quad 20 - 7 = 13
\][/tex]
RHS:
[tex]\[
\frac{3}{4} \cdot 8 = 6, \quad \text{so} \quad 6 + 14 = 20
\][/tex]
Clearly, [tex]\( 13 \neq 20 \)[/tex].
### Conclusion
The equation that results in [tex]\( x = 8 \)[/tex] is:
[tex]\[
C. \quad \frac{5}{4} x - 2 = \frac{3}{2} x - 4
\][/tex]
