To find the equation of a line that is parallel to [tex]\( y = 3x + 5 \)[/tex] and passes through the point [tex]\( (2, 5) \)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
The given equation of the line is [tex]\( y = 3x + 5 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope. Thus, the slope of the given line is [tex]\( 3 \)[/tex].
2. Determine the form of the parallel line's equation:
A line that is parallel to the given line will have the same slope. Therefore, the equation of our parallel line will have the form [tex]\( y = 3x + c \)[/tex], where [tex]\( c \)[/tex] is the y-intercept we need to find.
3. Use the given point to find the y-intercept:
The line we are looking for must pass through the point [tex]\( (2, 5) \)[/tex]. This means when [tex]\( x = 2 \)[/tex], [tex]\( y = 5 \)[/tex]. We can substitute these values into the equation [tex]\( y = 3x + c \)[/tex] to find [tex]\( c \)[/tex].
Substituting [tex]\( x = 2 \)[/tex] and [tex]\( y = 5 \)[/tex]:
[tex]\[ 5 = 3(2) + c \][/tex]
4. Solve for the y-intercept [tex]\( c \)[/tex]:
[tex]\[ 5 = 6 + c \][/tex]
Subtract 6 from both sides to solve for [tex]\( c \)[/tex]:
[tex]\[ 5 - 6 = c \][/tex]
[tex]\[ -1 = c \][/tex]
5. Write the final equation:
With the slope [tex]\( 3 \)[/tex] and y-intercept [tex]\( -1 \)[/tex], the equation of the line is:
[tex]\[ y = 3x - 1 \][/tex]
Therefore, the equation of the line parallel to [tex]\( y = 3x + 5 \)[/tex] and passing through the point [tex]\( (2, 5) \)[/tex] is:
[tex]\[ y = 3x - 1 \][/tex]
