Question

solve the equation.

• 3(4p + 5) - 2(2 - 11p) = 3(8 + 3p)

p = 9.5

Explanation:

Step1: Expand all terms

First, we use the distributive property (also known as the distributive law) to expand each of the parentheses. The distributive property states that \(a(b + c)=ab+ac\).

For the left - hand side:

• Expand \(-3(4p + 5)\): \(-3\times4p+(-3)\times5=-12p - 15\)
• Expand \(-2(2 - 11p)\): \(-2\times2+(-2)\times(-11p)=-4 + 22p\)

So the left - hand side of the equation becomes \((-12p-15)+(-4 + 22p)\). Combine like terms: \((-12p+22p)+(-15 - 4)=10p-19\)

For the right - hand side:
Expand \(3(8 + 3p)\): \(3\times8+3\times3p = 24+9p\)

Now our equation is \(10p-19=24 + 9p\)

Step2: Solve for \(p\)

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

Then add 19 to both sides:
\(p-19 + 19=24 + 19\)
Simplify: \(p=43\)? Wait, there is a mistake in the previous calculation. Let's re - do the expansion.

Wait, let's start over:

Original equation: \(-3(4p + 5)-2(2 - 11p)=3(8 + 3p)\)

Expand left - hand side:

\(-3\times4p-3\times5-2\times2+2\times11p=-12p-15 - 4+22p\)

Combine like terms: \((-12p + 22p)+(-15-4)=10p-19\)

Right - hand side: \(3\times8+3\times3p = 24 + 9p\)

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

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

We get \(p-19 = 24\)

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

Wait, the given answer is \(p = 9.5\), which is 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:

\(-3(4p + 5)-2(2 - 11p)=-12p-15-4 + 22p\) (because \(-2\times2=-4\) and \(-2\times(-11p)=22p\), that part is correct)

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

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

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

Now, move all \(p\) terms to left and constants to right:

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

\(p=43\)

So the correct value of \(p\) is \(43\), not \(9.5\). Let's verify:

Left - hand side: \(-3(4\times43+5)-2(2 - 11\times43)\)

First, \(4\times43 = 172\), \(172 + 5=177\), \(-3\times177=-531\)

\(2-11\times43=2 - 473=-471\), \(-2\times(-471)=942\)

Left - hand side: \(-531+942 = 411\)

Right - hand side: \(3(8 + 3\times43)=3(8 + 129)=3\times137 = 411\)

So \(p = 43\) is correct.

Answer:

\(p = 43\)