Question

question
solve this equation for p:
\\(\frac{p + 1}{2} = \frac{3p + 7}{5}\\).
type the correct answer in the box. use numerals instead of words.
\\(p = \square\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

To solve the equation \(\frac{p + 1}{2}=\frac{3p+7}{5}\), we cross - multiply. Cross - multiplying gives us \(5(p + 1)=2(3p + 7)\). The reason for cross - multiplying is that if \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(ad = bc\).

Step2: Expand both sides of the equation

Expanding the left - hand side: \(5(p + 1)=5p+5\) (using the distributive property \(a(b + c)=ab+ac\) with \(a = 5\), \(b=p\), and \(c = 1\)).
Expanding the right - hand side: \(2(3p + 7)=6p+14\) (using the distributive property \(a(b + c)=ab + ac\) with \(a = 2\), \(b = 3p\), and \(c=7\)).
So our equation becomes \(5p + 5=6p+14\).

Step3: Isolate the variable \(p\)

Subtract \(5p\) from both sides of the equation:
\(5p+5-5p=6p + 14-5p\)
Simplifying both sides, we get \(5=p + 14\).
Then subtract 14 from both sides: \(5-14=p+14 - 14\).
Simplifying, we have \(- 9=p\).

Answer:

\(-9\)