Certainly! Let's solve the equation step by step:
We start with the given equation:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
Our goal is to solve for [tex]\( x \)[/tex].
### Step 1: Distribute [tex]\( m \)[/tex]
We can distribute [tex]\( m \)[/tex] on the right-hand side:
[tex]\[ y - y_1 = m x - m x_1 \][/tex]
### Step 2: Isolate the term involving [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], we add [tex]\( m x_1 \)[/tex] to both sides of the equation:
[tex]\[ y - y_1 + m x_1 = m x \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Now, we divide both sides by [tex]\( m \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{y - y_1 + m x_1}{m} \][/tex]
This can be simplified as:
[tex]\[ x = \frac{y - y_1}{m} + \frac{m x_1}{m} \][/tex]
Since [tex]\(\frac{m x_1}{m}\)[/tex] simplifies to [tex]\( x_1 \)[/tex], we get:
[tex]\[ x = \frac{y - y_1}{m} + x_1 \][/tex]
### Conclusion
Now, let's compare this solution with the given options:
A. [tex]\( x = \frac{y - y_1 + x_1}{m} \)[/tex]
B. [tex]\( x = \frac{y - y_1}{m} - x_1 \)[/tex]
C. [tex]\( x = \frac{y - y_1}{m} + x_1 \)[/tex]
D. [tex]\( x = \frac{m \left(y - y_1\right)}{x_1} \)[/tex]
The correct option that matches our simplified solution is:
[tex]\[ \boxed{C} \][/tex]
Thus, the correct solution is [tex]\( x = \frac{y - y_1}{m} + x_1 \)[/tex], which corresponds to option C.
