To solve the equation [tex]\( \frac{p+1}{2} = \frac{3p+7}{5} \)[/tex] for [tex]\( p \)[/tex], follow these steps:
1. Eliminate the fractions by finding a common denominator. The denominators here are 2 and 5, and their least common multiple is 10. Multiply both sides of the equation by 10:
[tex]\[
10 \cdot \frac{p+1}{2} = 10 \cdot \frac{3p+7}{5}
\][/tex]
2. Simplify the resulting equation:
[tex]\[
5(p + 1) = 2(3p + 7)
\][/tex]
3. Distribute both sides:
[tex]\[
5p + 5 = 6p + 14
\][/tex]
4. Isolate the variable [tex]\( p \)[/tex]. Start by getting all the terms involving [tex]\( p \)[/tex] on one side and the constant terms on the other side. Subtract [tex]\( 5p \)[/tex] from both sides:
[tex]\[
5 = p + 14
\][/tex]
5. Solve for [tex]\( p \)[/tex] by subtracting 14 from both sides:
[tex]\[
p = -9
\][/tex]
The solution to the equation is:
[tex]\[
p = -9
\][/tex]
