Determine x so that the slope of the line through (1, 4) and (x, 2) is 2. 

Answer

To determine the value of \( x \) such that the slope of the line passing through the points \( (1, 4) \) and \( (x, 2) \) is equal to 2, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the points and the slope formula**:
We have two points: \( (x_1, y_1) = (1, 4) \) and \( (x_2, y_2) = (x, 2) \). The formula for the slope \( m \) between two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

2. **Substitute the known values into the slope formula**:
We know the slope \( m = 2 \). Substituting the values into the slope formula, we get:
\[ 2 = \frac{2 - 4}{x - 1} \]

3. **Simplify the equation**:
The numerator simplifies to \( 2 - 4 = -2 \). Thus, we have:
\[ 2 = \frac{-2}{x - 1} \]

4. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives:
\[ 2(x - 1) = -2 \]

5. **Distribute on the left side**:
Distributing \( 2 \) yields:
\[ 2x - 2 = -2 \]

6. **Add 2 to both sides**:
To isolate the term with \( x \), add \( 2 \) to both sides:
\[ 2x - 2 + 2 = -2 + 2 \]
This simplifies to:
\[ 2x = 0 \]

7. **Divide by 2 to solve for \( x \)**:
Finally, divide both sides by \( 2 \):
\[ x = 0 \]

### Final Answer:
The value of \( x \) is \( 0 \).

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