If (3 p upon 7+7upon6p-(3pupon 7-7p upon 6 =r find the value of of constant r

Let's simplify the expression step by step to find the value of the constant \( r \).

Given expression:

\[
\frac{3p}{7} + \frac{7}{6p} - \left(\frac{3p}{7} - \frac{7p}{6}\right) = r
\]

### Step 1: Simplify the expression inside the parentheses

\[
\left(\frac{3p}{7} - \frac{7p}{6}\right)
\]

To simplify, find a common denominator:

The least common multiple (LCM) of 7 and 6 is 42.

\[
\frac{3p}{7} = \frac{3p \times 6}{42} = \frac{18p}{42}
\]
\[
\frac{7p}{6} = \frac{7p \times 7}{42} = \frac{49p}{42}
\]

So, 

\[
\frac{3p}{7} - \frac{7p}{6} = \frac{18p}{42} - \frac{49p}{42} = \frac{18p - 49p}{42} = \frac{-31p}{42}
\]

### Step 2: Substitute back into the original equation

Now, the equation becomes:

\[
\frac{3p}{7} + \frac{7}{6p} - \left(\frac{-31p}{42}\right) = r
\]

The minus sign before the parentheses will change the sign of the expression inside:

\[
\frac{3p}{7} + \frac{7}{6p} + \frac{31p}{42} = r
\]

### Step 3: Simplify the expression

To simplify the expression, we need to find a common denominator for the fractions:

1. The LCM of 7 and 42 is 42.

\[
\frac{3p}{7} = \frac{3p \times 6}{42} = \frac{18p}{42}
\]

So, the expression becomes:

\[
\frac{18p}{42} + \frac{7}{6p} + \frac{31p}{42} = \frac{18p + 31p}{42} + \frac{7}{6p} = \frac{49p}{42} + \frac{7}{6p} = r
\]

2. Combine the fractions:

\[
\frac{49p}{42} + \frac{7}{6p} = r
\]

So, the value of \( r \) is:

\[
r = \frac{49p}{42} + \frac{7}{6p}
\]

This is the simplified form of \( r \).

Would you like a more detailed explanation or help with anything else?

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Here are 8 related questions to explore further:

1. How do you find the least common denominator for multiple fractions?
2. What happens if the variable \( p \) is equal to zero in this equation?
3. How would the value of \( r \) change if \( p \) is a negative number?
4. How can you simplify fractions that contain variables in the denominator?
5. How would you solve this equation if it included trigonometric functions?
6. What methods can be used to check if the simplified form of \( r \) is correct?
7. How would you solve a similar problem if the terms involved square roots?
8. How do you handle equations involving complex numbers?

**Tip:** Always cross-check your work by substituting simplified results back into the original equation.

Learn how to solve an expression involving fractional terms and constants. This problem covers simplifying expressions with fractions and finding the value of a constant. Suitable for students in Grades 7-9.

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