Question

1. for what value of p makes the following equation true?

\\(\frac{p - 4}{7} = \frac{2p + 1}{5}\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

To solve the equation \(\frac{p - 4}{7}=\frac{2p+1}{5}\), we cross - multiply. Cross - multiplying gives us \(5(p - 4)=7(2p + 1)\). The reason for cross - multiplying is that if \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d=b\times c\).

Step2: Expand both sides

Expand the left - hand side: \(5(p - 4)=5p-20\) (using the distributive property \(a(b - c)=ab - ac\) where \(a = 5\), \(b=p\), and \(c = 4\)).
Expand the right - hand side: \(7(2p + 1)=14p+7\) (using the distributive property \(a(b + c)=ab+ac\) where \(a = 7\), \(b = 2p\), and \(c = 1\)).
So our equation becomes \(5p-20 = 14p+7\).

Step3: Move all \(p\) terms to one side and constants to the other

Subtract \(5p\) from both sides: \(5p-5p - 20=14p-5p + 7\), which simplifies to \(-20 = 9p+7\).
Then subtract 7 from both sides: \(-20 - 7=9p+7 - 7\), which gives \(-27 = 9p\).

Step4: Solve for \(p\)

Divide both sides by 9: \(\frac{-27}{9}=\frac{9p}{9}\).
Simplifying, we get \(p=- 3\).

Answer:

\(p=-3\)