Solve : `2(x-5) +3(x-2) =8+7(x-4)` and check your answer.

Answer

To solve the equation \(2(x-5) + 3(x-2) = 8 + 7(x-4)\), we will follow these steps:

### Step 1: Expand both sides of the equation.
We start by distributing the terms on both sides of the equation.

\[ 2(x - 5) + 3(x - 2) = 8 + 7(x - 4) \]

Expanding the left side:
\[ 2x - 10 + 3x - 6 \]

Expanding the right side:
\[ 8 + 7x - 28 \]

### Step 2: Combine like terms.
Now, we combine the like terms on both sides.

Left side:
\[ (2x + 3x) + (-10 - 6) = 5x - 16 \]

Right side:
\[ 8 - 28 + 7x = 7x - 20 \]

So, the equation now looks like:
\[ 5x - 16 = 7x - 20 \]

### Step 3: Rearrange the equation.
Next, we will move all terms involving \(x\) to one side and constant terms to the other side.

Subtract \(7x\) from both sides:
\[ 5x - 7x - 16 = -20 \]

This simplifies to:
\[ -2x - 16 = -20 \]

Now, add 16 to both sides:
\[ -2x = -20 + 16 \]

This simplifies to:
\[ -2x = -4 \]

### Step 4: Solve for \(x\).
Now, divide both sides by -2 to find \(x\):
\[ x = \frac{-4}{-2} = 2 \]

### Step 5: Check the solution.
To verify our solution, we substitute \(x = 2\) back into the original equation.

Left side:
\[ 2(2 - 5) + 3(2 - 2) = 2(-3) + 3(0) = -6 + 0 = -6 \]

Right side:
\[ 8 + 7(2 - 4) = 8 + 7(-2) = 8 - 14 = -6 \]

Since both sides are equal, our solution is verified.

### Final Answer:
\[ x = 2 \]