To find the equations of the lines that are parallel to the line given by the equation y = 3x + 5, we first need to understand the characteristics of parallel lines in geometry.
Parallel lines have the same slope but different y-intercepts. The equation of the line is expressed in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
In our case, the slope m of the line y = 3x + 5 is 3, and the y-intercept b is 5.
To form equations of lines that are parallel to this line, we will keep the slope as 3 but change the y-intercept. This gives us a family of equations of the form:
- y = 3x + b
Here, b can take any real number value. For example:
- If b = 0, the equation would be y = 3x.
- If b = 1, the equation would be y = 3x + 1.
- If b = -2, the equation would be y = 3x – 2.
Thus, any line in the form of y = 3x + b, where b is any real number, represents a line parallel to the original line.