Certainly! Let's solve this step-by-step to find the equation of a line that is parallel to the given line and passes through the specified point.
We start with the given line:
[tex]\[ y = 3x + 2 \][/tex]
### Step 1: Determine the Slope
The slope of a line in the form [tex]\( y = mx + b \)[/tex] is [tex]\( m \)[/tex], where [tex]\( m \)[/tex] represents the slope.
For the line [tex]\( y = 3x + 2 \)[/tex], the slope [tex]\( m \)[/tex] is:
[tex]\[ m = 3 \][/tex]
### Step 2: Establish the General Form of the Parallel Line
Lines that are parallel have identical slopes. Therefore, the parallel line we seek will also have a slope of [tex]\( 3 \)[/tex]. The general form of this line is:
[tex]\[ y = 3x + c \][/tex]
where [tex]\( c \)[/tex] is the y-intercept that we need to find.
### Step 3: Use the Given Point
We are given the point through which this line passes: [tex]\( (10, 1) \)[/tex].
### Step 4: Substitute the Point into the Equation
To find [tex]\( c \)[/tex], substitute the coordinates of the point [tex]\( (10, 1) \)[/tex] into the general form of the equation [tex]\( y = 3x + c \)[/tex]:
[tex]\[ 1 = 3(10) + c \][/tex]
### Step 5: Solve for [tex]\( c \)[/tex]
[tex]\[
1 = 30 + c \\
c = 1 - 30 \\
c = -29
\][/tex]
### Step 6: Write the Equation of the Line
Now that we have found [tex]\( c \)[/tex], we can write the equation of the line:
[tex]\[ y = 3x - 29 \][/tex]
### Step 7: Convert to Standard Form
To express this equation in standard form (Ax + By = C), we rearrange it:
[tex]\[
y = 3x - 29 \\
3x - y = 29
\][/tex]
### Conclusion
The required equation of the line is:
[tex]\[ 3x - y = 29 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
