To solve the equation [tex]\(\frac{7}{x} - \frac{3}{2} = \frac{n - mx}{px}\)[/tex] and determine the values for [tex]\(m\)[/tex], [tex]\(n\)[/tex], and [tex]\(p\)[/tex], we need to express both terms on the left-hand side with a common denominator. Here's a step-by-step solution:
1. Start with the given equation:
[tex]\[
\frac{7}{x} - \frac{3}{2} = \frac{n - mx}{px}
\][/tex]
2. Identify a common denominator for the left-hand side of the equation. The common denominator for [tex]\(x\)[/tex] and 2 is [tex]\(2x\)[/tex]:
[tex]\[
\frac{7}{x} - \frac{3}{2} = \frac{7 \cdot 2}{2x} - \frac{3 \cdot x}{2x}
\][/tex]
3. Rewrite each term with the common denominator [tex]\(2x\)[/tex]:
[tex]\[
\frac{14}{2x} - \frac{3x}{2x} = \frac{14 - 3x}{2x}
\][/tex]
4. Combine the terms on the left-hand side:
[tex]\[
\frac{14 - 3x}{2x}
\][/tex]
5. Compare the left-hand side of the equation with the right-hand side:
[tex]\[
\frac{14 - 3x}{2x} = \frac{n - mx}{px}
\][/tex]
6. From the comparison, it is clear that:
[tex]\[
n = 14, \quad m = 3, \quad p = 2
\][/tex]
Thus, the values for [tex]\(m\)[/tex], [tex]\(n\)[/tex], and [tex]\(p\)[/tex] that complete the difference are:
[tex]\[
m = 3
\][/tex]
[tex]\[
n = 14
\][/tex]
[tex]\[
p = 2
\][/tex]
