Before we begin, please note that there are infinitely many possibilities that would satisfy the required conditions. Here we will look at only one scenario.
We will write the equations of the lines in slope-intercept form.
y=mx+ b
In this form, m represents the slope of the line and b represents the y-intercept. Let's think of the slopes of our lines first, and then we will complete the equation by choosing the y-intercepts.
Slopes
We know that when two lines are perpendicular, then the product of their slopes is -1.
m_1*m_2=-1
Let's say that m_1=2. To find m_2, we can substitute 2 for m_1 in our equation.
Now that we found the possible slopes of our lines, we can write the following equations of our lines.
Line 1:&y=2x+ b_1
Line 2:&y=- 12x+ b_2
y-intercepts
We want the lines to have the same y-intercept. However, we also do not want them to pass through the origin. If they passed through the origin, then their y-intercepts would be 0. Thus, we can choose any value for b_1 and b_2, but not 0. Let's take b_1=b_2=3. Finally, we can complete our equations.
Line 1:&y=2x+ 3
Line 2:&y=- 12x+ 3
As we can see, the slopes are opposite reciprocals, so they are perpendicular. The y-intercepts are the same, so the equations satisfy the conditions from the exercise.
