Question

solve the given equation.
1.) $4x - 6 = x + 9$
2.) $4 - 7x = 1 - 6x$
5.) $6(2 + y) = 3(3 - y)$
6.) $4y = 2(y - 5) - 2$
7.) $6x - 9x - 4 = -2x - 2$
8.) $-9x + 6 = -x + 4$
11.) $3x - 4 + 5x + 14 = 90$
12.) $5t + 13 + t + 23 = 180$

Explanation:

Let's solve each equation one by one:

1) Solve \( 4x - 6 = x + 9 \)

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

To get all \( x \)-terms on one side, we subtract \( x \) from both sides:
\( 4x - x - 6 = x - x + 9 \)
\( 3x - 6 = 9 \)

Step 2: Add 6 to both sides

To isolate the term with \( x \), we add 6 to both sides:
\( 3x - 6 + 6 = 9 + 6 \)
\( 3x = 15 \)

Step 3: Divide by 3

To solve for \( x \), we divide both sides by 3:
\( \frac{3x}{3} = \frac{15}{3} \)
\( x = 5 \)

Step 1: Add \( 7x \) to both sides

To get all \( x \)-terms on one side, we add \( 7x \) to both sides:
\( 4 - 7x + 7x = 1 - 6x + 7x \)
\( 4 = 1 + x \)

Step 2: Subtract 1 from both sides

To isolate \( x \), we subtract 1 from both sides:
\( 4 - 1 = 1 - 1 + x \)
\( 3 = x \)

Step 1: Distribute both sides

First, we distribute the 6 and 3:
\( 6 \times 2 + 6y = 3 \times 3 - 3y \)
\( 12 + 6y = 9 - 3y \)

Step 2: Add \( 3y \) to both sides

To get all \( y \)-terms on one side, we add \( 3y \) to both sides:
\( 12 + 6y + 3y = 9 - 3y + 3y \)
\( 12 + 9y = 9 \)

Step 3: Subtract 12 from both sides

To isolate the term with \( y \), we subtract 12 from both sides:
\( 12 - 12 + 9y = 9 - 12 \)
\( 9y = -3 \)

Step 4: Divide by 9

To solve for \( y \), we divide both sides by 9:
\( \frac{9y}{9} = \frac{-3}{9} \)
\( y = -\frac{1}{3} \)

Answer:

\( x = 5 \)

2) Solve \( 4 - 7x = 1 - 6x \)