Certainly! To find the equations of lines that are parallel to the given lines and passing through specific points, we need to use the concept of slope and the point-slope form of a line. Parallel lines have the same slope.
a) We are given a line with equation y = 5x + 1, which is already in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. For this line, the slope (m) is 5.
Since parallel lines have the same slope, the line we want to find must also have a slope of 5. This new line passes through the point (0, 4), which actually gives us the y-intercept directly because when x is 0, y is 4 for this point. Therefore, the y-intercept (b) of our new line is 4. We can write the equation of this line in slope-intercept form:
y = mx + b
y = 5x + 4
This is the equation of line a.
b) Now, we are given another line with equation y = 4x - 9, which also is in slope-intercept form. For this line, the slope (m) is 4.
Again, since parallel lines have the same slope, our new line must have a slope of 4. The point this line passes through is (0, -1), so when x is 0, y is -1. This gives us the y-intercept of the new line, which is -1.
Using the slope-intercept form again:
y = mx + b
y = 4x - 1
This is the equation of line b.
In summary, the equations of the lines are:
a) y = 5x + 4, which is parallel to y = 5x + 1 and passes through (0, 4).
b) y = 4x - 1, which is parallel to y = 4x - 9 and passes through (0, -1).
