Equation of the line passing through `(1,2)` and parallel to the line `y=3x-1` is

Options

A. `y+2=x+1`
B. `y+2=3(x+1)`
C. `y-2=3(x-1)`
D. `y-2=x-1`

Explanation

To find the equation of the line that passes through the point (1, 2) and is parallel to the line given by the equation \( y = 3x - 1 \), we can follow these steps:

### Step 1: Identify the slope of the given line
The equation of the line is given in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. From the equation \( y = 3x - 1 \), we can see that the slope \( m \) is 3.

**Hint:** The slope-intercept form of a line is \( y = mx + b \). The coefficient of \( x \) gives the slope.

### Step 2: Use the slope for the new line
Since the required line is parallel to the given line, it will have the same slope. Therefore, the slope of the required line is also 3.

**Hint:** Parallel lines have the same slope.

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of a line is given by the formula:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Here, \( (x_1, y_1) = (1, 2) \) and \( m = 3 \).

**Hint:** The point-slope form is useful when you know a point on the line and the slope.

### Step 4: Substitute the values into the point-slope formula
Substituting \( m = 3 \), \( x_1 = 1 \), and \( y_1 = 2 \) into the point-slope formula, we get:
\[ y - 2 = 3(x - 1) \]

**Hint:** Make sure to substitute the correct values for \( x_1 \), \( y_1 \), and \( m \).

### Step 5: Simplify the equation
Now, we will simplify the equation:
\[ y - 2 = 3(x - 1) \]
Expanding the right side:
\[ y - 2 = 3x - 3 \]
Now, add 2 to both sides:
\[ y = 3x - 3 + 2 \]
\[ y = 3x - 1 \]

**Hint:** Always keep track of your arithmetic when simplifying.

### Final Answer
The equation of the line passing through (1, 2) and parallel to the line \( y = 3x - 1 \) is:
\[ y = 3x - 1 \]