Solve the following equations and verify the results.
`(x-6)/4-(x-4)/6=1-x/10`

Answer

To solve the equation \((x-6)/4 - (x-4)/6 = 1 - x/10\), we will follow these steps:

### Step 1: Find a common denominator
The denominators in the equation are 4, 6, and 10. The least common multiple (LCM) of these numbers is 60. We will multiply the entire equation by 60 to eliminate the fractions.

\[ 60 \left(\frac{x-6}{4}\right) - 60 \left(\frac{x-4}{6}\right) = 60 \left(1 - \frac{x}{10}\right) \]

### Step 2: Simplify each term
Now we simplify each term:

\[ 15(x-6) - 10(x-4) = 60 - 6x \]

### Step 3: Distribute
Next, we distribute the constants in front of the parentheses:

\[ 15x - 90 - 10x + 40 = 60 - 6x \]

### Step 4: Combine like terms
Now, we combine like terms on the left side:

\[ (15x - 10x) + (-90 + 40) = 60 - 6x \]

This simplifies to:

\[ 5x - 50 = 60 - 6x \]

### Step 5: Move all \(x\) terms to one side
We will add \(6x\) to both sides to get all \(x\) terms on one side:

\[ 5x + 6x - 50 = 60 \]

This simplifies to:

\[ 11x - 50 = 60 \]

### Step 6: Move constants to the other side
Next, we add 50 to both sides:

\[ 11x = 110 \]

### Step 7: Solve for \(x\)
Now, we divide both sides by 11:

\[ x = 10 \]

### Step 8: Verify the solution
To verify, we substitute \(x = 10\) back into the original equation:

\[ \frac{10-6}{4} - \frac{10-4}{6} = 1 - \frac{10}{10} \]

Calculating each side:

Left side:

\[ \frac{4}{4} - \frac{6}{6} = 1 - 1 = 0 \]

Right side:

\[ 0 = 0 \]

Since both sides are equal, our solution is verified.

### Final Answer:
\[ x = 10 \]