To find the equation of the line that is parallel to [tex]\( y = 4x + 1 \)[/tex] and passes through the point [tex]\((1, 1)\)[/tex], follow these steps:
1. Identify the slope of the given line:
- The given line's equation is [tex]\( y = 4x + 1 \)[/tex].
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- Therefore, the slope ([tex]\( m \)[/tex]) of the given line is [tex]\( 4 \)[/tex].
2. Determine the slope of the parallel line:
- Parallel lines have the same slope.
- Consequently, the slope of the line we are looking for is also [tex]\( 4 \)[/tex].
3. Use the point-slope form to write the equation of the line:
- The point-slope form of a line’s equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\( m \)[/tex] is the slope.
- Here, the point [tex]\((1, 1)\)[/tex] lies on the line, so [tex]\( x_1 = 1 \)[/tex] and [tex]\( y_1 = 1 \)[/tex].
4. Substitute the slope and the point into the point-slope form equation:
- We have [tex]\( y - 1 = 4(x - 1) \)[/tex].
5. Simplify this equation to get it into slope-intercept form [tex]\( y = mx + b \)[/tex]:
- Distribute the slope [tex]\( 4 \)[/tex]:
[tex]\[ y - 1 = 4(x - 1) \][/tex]
[tex]\[ y - 1 = 4x - 4 \][/tex]
- Add 1 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 4 + 1 \][/tex]
[tex]\[ y = 4x - 3 \][/tex]
6. Write the final equation of the line:
- The equation of the line parallel to [tex]\( y = 4x + 1 \)[/tex] and passing through [tex]\((1, 1)\)[/tex] is:
[tex]\[ y = 4x - 3 \][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = 4x + 1 \)[/tex] and contains the point [tex]\( (1, 1) \)[/tex] is [tex]\( y = 4x - 3 \)[/tex].
