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Which equation, when solved, gives 8 for the value of [tex] x [/tex]?
A. [tex] \frac{5}{2} x + \frac{7}{2} = \frac{3}{4} x + 14 [/tex]
B. [tex] \frac{5}{4} x - 9 = \frac{3}{2} x - 12 [/tex]
C. [tex] \frac{5}{4} x - 2 = \frac{3}{2} x - 4 [/tex]
D. [tex] \frac{5}{2} x - 7 = \frac{3}{4} x + 14 [/tex]

Sagot :

GinnyAnswer
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]
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