To solve the equation [tex]\(\frac{p+1}{2} = \frac{3p+7}{5}\)[/tex], we will go through the following steps:
1. Eliminate the fractions by finding a common multiple of the denominators (2 and 5), which is 10. Multiply both sides of the equation by 10 to clear the denominators.
[tex]\[
10 \cdot \frac{p+1}{2} = 10 \cdot \frac{3p+7}{5}
\][/tex]
2. Simplify the equation after multiplying:
[tex]\[
5(p+1) = 2(3p+7)
\][/tex]
3. Distribute the numbers on both sides of the equation:
[tex]\[
5p + 5 = 6p + 14
\][/tex]
4. To isolate [tex]\(p\)[/tex], move all the [tex]\(p\)[/tex]-terms to one side and the constant terms to the other side. Start by subtracting [tex]\(5p\)[/tex] from both sides:
[tex]\[
5p + 5 - 5p = 6p + 14 - 5p
\][/tex]
Simplifying this, we get:
[tex]\[
5 = p + 14
\][/tex]
5. Next, subtract 14 from both sides to solve for [tex]\(p\)[/tex]:
[tex]\[
5 - 14 = p + 14 - 14
\][/tex]
Simplifying this, we get:
[tex]\[
-9 = p
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{p + 1}{2} = \frac{3p + 7}{5}\)[/tex] is:
[tex]\[
p = -9
\][/tex]
