Question

1. 11x - 2 = 16 + 7x
2. 24 + 24n - 10n = -6n + 24
3. 40x + 10 - 25x - 5 = -12x + 15
4. 3x + 7 + 5x = 2x + 30 + 40
5. 12x - 14 = 11

Explanation:

Problem 1: Solve \( 11x - 2 = 16 + 7x \)

Step 1: Subtract \( 7x \) from both sides

To isolate the \( x \)-terms on one side, we subtract \( 7x \) from both sides of the equation. This gives us:
\( 11x - 7x - 2 = 16 + 7x - 7x \)
Simplifying the left and right sides:
\( 4x - 2 = 16 \)

Step 2: Add 2 to both sides

To isolate the term with \( x \), we add 2 to both sides of the equation:
\( 4x - 2 + 2 = 16 + 2 \)
Simplifying both sides:
\( 4x = 18 \)

Step 3: Divide by 4

To solve for \( x \), we divide both sides of the equation by 4:
\( \frac{4x}{4} = \frac{18}{4} \)
Simplifying:
\( x = \frac{9}{2} = 4.5 \)

Problem 2: Solve \( 24 + 24n - 10n = -6n + 24 \)

Step 1: Combine like terms

First, combine the \( n \)-terms on the left side:
\( 24 + (24n - 10n) = -6n + 24 \)
Simplifying the left side:
\( 24 + 14n = -6n + 24 \)

Step 2: Add \( 6n \) to both sides

To get all \( n \)-terms on the left, add \( 6n \) to both sides:
\( 24 + 14n + 6n = -6n + 6n + 24 \)
Simplifying:
\( 24 + 20n = 24 \)

Step 3: Subtract 24 from both sides

Subtract 24 from both sides to isolate the \( n \)-term:
\( 24 - 24 + 20n = 24 - 24 \)
Simplifying:
\( 20n = 0 \)

Step 4: Divide by 20

Divide both sides by 20:
\( \frac{20n}{20} = \frac{0}{20} \)
Simplifying:
\( n = 0 \)

Problem 3: Solve \( 40x + 10 - 25x - 5 = -12x + 15 \)

Step 1: Combine like terms

Combine the \( x \)-terms and constant terms on the left side:
\( (40x - 25x) + (10 - 5) = -12x + 15 \)
Simplifying:
\( 15x + 5 = -12x + 15 \)

Step 2: Add \( 12x \) to both sides

Add \( 12x \) to both sides to get all \( x \)-terms on the left:
\( 15x + 12x + 5 = -12x + 12x + 15 \)
Simplifying:
\( 27x + 5 = 15 \)

Step 3: Subtract 5 from both sides

Subtract 5 from both sides to isolate the \( x \)-term:
\( 27x + 5 - 5 = 15 - 5 \)
Simplifying:
\( 27x = 10 \)

Step 4: Divide by 27

Divide both sides by 27:
\( \frac{27x}{27} = \frac{10}{27} \)
Simplifying:
\( x = \frac{10}{27} \)

Problem 4: Solve \( 3x + 7 + 5x = 2x + 30 + 40 \)

Answer:

s:

1. \( x = 4.5 \) (or \( \frac{9}{2} \))
2. \( n = 0 \)
3. \( x = \frac{10}{27} \)
4. \( x = 10.5 \) (or \( \frac{21}{2} \))
5. \( x = \frac{25}{12} \) (or \( \approx 2.083 \))