Question

solve the equation.
-3(4p + 5) - 2(2 - 11p) = 3(8 + 3p)

p = 9.5

Explanation:

Step1: Expand all terms

Using the distributive property \(a(b + c)=ab+ac\), we expand each parenthesis:
\(-3(4p + 5)=-12p-15\), \(-2(2 - 11p)=-4 + 22p\), and \(3(8 + 3p)=24+9p\).
So the equation becomes \(-12p - 15-4 + 22p=24 + 9p\).

Step2: Combine like terms

On the left - hand side, combine the \(p\) terms and the constant terms:
\((-12p+22p)+(-15 - 4)=10p-19\).
The equation is now \(10p-19 = 24+9p\).

Step3: Isolate the variable \(p\)

Subtract \(9p\) from both sides:
\(10p-9p-19=24 + 9p-9p\), which simplifies to \(p-19 = 24\).
Then add 19 to both sides:
\(p-19 + 19=24 + 19\), so \(p=43\)? Wait, there was a mistake in the previous calculation. Let's re - do the expansion:

Correct Step1: Expand the terms
\(-3(4p + 5)=-12p-15\)
\(-2(2 - 11p)=-4 + 22p\)
\(3(8 + 3p)=24+9p\)

So the left - hand side: \(-12p-15-4 + 22p=(-12p + 22p)+(-15-4)=10p-19\)

The equation: \(10p-19=24 + 9p\)

Subtract \(9p\) from both sides: \(10p-9p-19=24+9p - 9p\) gives \(p-19 = 24\)

Add 19 to both sides: \(p=24 + 19=43\)

Wait, the original answer in the box was 9.5, which was wrong. Let's check the expansion again.

Wait, maybe I made a mistake in the sign when expanding \(-2(2 - 11p)\). Let's re - expand:

\(-2\times2=-4\) and \(-2\times(-11p)=22p\), that part is correct.

\(-3(4p + 5)=-12p-15\), correct.

\(3(8 + 3p)=24 + 9p\), correct.

Left - hand side: \(-12p-15-4 + 22p=10p-19\)

Right - hand side: \(24 + 9p\)

So \(10p-19=24 + 9p\)

Subtract \(9p\): \(p-19 = 24\)

Add 19: \(p=43\)

Wait, the initial boxed answer was 9.5, which is incorrect. Let's solve the equation again carefully:

Starting with \(-3(4p + 5)-2(2 - 11p)=3(8 + 3p)\)

Expand each term:

\(-12p-15-4 + 22p=24+9p\)

Combine like terms on the left:

\((-12p + 22p)+(-15-4)=10p-19\)

So \(10p-19=24 + 9p\)

Subtract \(9p\) from both sides: \(10p-9p-19=24+9p - 9p\)

\(p-19 = 24\)

Add 19 to both sides: \(p=24 + 19=43\)

Answer:

\(p = 43\)