The point (x,y) lies on the line with slope m and passes through the fixed point `(x_(0),y_(0))` if and only if its coordinates satisfy the equation `y-y_(0)` is equal to

Options

A. `m(x-x_(0))`
B. `m(y-x_(0))`
C. `m(y-x)`
D. `m(x-y_(0))`

Answer

To solve the problem, we need to derive the equation of a line that has a slope \( m \) and passes through a fixed point \( (x_0, y_0) \).

### Step-by-Step Solution:

1. **Understanding the Slope**:
The slope \( m \) of a line is defined as the ratio of the change in \( y \) to the change in \( x \). Mathematically, this can be expressed as:
\[ m = \frac{y - y_0}{x - x_0} \]
where \( (x_0, y_0) \) is a point on the line and \( (x, y) \) is any other point on the line.

**Hint**: Recall that the slope is the rise over run, which means how much \( y \) changes for a given change in \( x \).

2. **Rearranging the Equation**:
We can rearrange the equation to isolate \( y - y_0 \):
\[ y - y_0 = m(x - x_0) \]

**Hint**: When rearranging equations, ensure you maintain equality by performing the same operation on both sides.

3. **Final Equation**:
The equation \( y - y_0 = m(x - x_0) \) represents the line with slope \( m \) passing through the point \( (x_0, y_0) \). This is the point-slope form of the equation of a line.

**Hint**: Recognize that this form is useful for writing the equation of a line when you know a point on the line and the slope.

### Conclusion:
The coordinates \( (x, y) \) satisfy the equation:
\[ y - y_0 = m(x - x_0) \]
This means that any point \( (x, y) \) on the line will satisfy this equation.