Question

solve the following equations. check your solution.

1. \\( 36 = 3(b + 3) + 9 \\)
2. \\( 9p + 4 - p - 20 = 56 \\)

Explanation:

Problem 65: Solve \( 36 = 3(b + 3) + 9 \) and check the solution.

Step 1: Simplify the right - hand side

First, we simplify the expression \( 3(b + 3)+9 \). Using the distributive property \( a(b + c)=ab+ac \), we have \( 3(b + 3)=3b+9 \). Then the equation becomes:
\( 36=3b + 9+9 \)
Simplify the constants on the right - hand side: \( 9 + 9=18 \), so the equation is \( 36=3b+18 \)

Step 2: Isolate the term with \( b \)

Subtract 18 from both sides of the equation to get rid of the constant on the right - hand side.
\( 36-18=3b+18 - 18 \)
\( 18 = 3b \)

Step 3: Solve for \( b \)

Divide both sides of the equation by 3.
\( \frac{18}{3}=\frac{3b}{3} \)
\( 6 = b \) or \( b = 6 \)

Step 4: Check the solution

Substitute \( b = 6 \) into the original equation \( 36=3(b + 3)+9 \)
Left - hand side: \( 36 \)
Right - hand side: \( 3(6 + 3)+9=3\times9+9 = 27 + 9=36 \)
Since the left - hand side equals the right - hand side, \( b = 6 \) is the correct solution.

Problem 66: Solve \( 9p+4 - p-20 = 56 \) and check the solution.

Step 1: Combine like terms

Combine the \( p \) terms and the constant terms on the left - hand side.
For the \( p \) terms: \( 9p-p = 8p \)
For the constant terms: \( 4-20=-16 \)
So the equation becomes \( 8p-16 = 56 \)

Step 2: Isolate the term with \( p \)

Add 16 to both sides of the equation.
\( 8p-16 + 16=56 + 16 \)
\( 8p=72 \)

Step 3: Solve for \( p \)

Divide both sides of the equation by 8.
\( \frac{8p}{8}=\frac{72}{8} \)
\( p = 9 \)

Step 4: Check the solution

Substitute \( p = 9 \) into the original equation \( 9p+4 - p-20 = 56 \)
Left - hand side: \( 9\times9+4-9 - 20=81 + 4-9 - 20=85-9 - 20=76 - 20 = 56 \)
Right - hand side: \( 56 \)
Since the left - hand side equals the right - hand side, \( p = 9 \) is the correct solution.

Answer:

s:
For problem 65: \( b=\boxed{6} \)
For problem 66: \( p=\boxed{9} \)