Answer:
1. Yes, the lines are perpendicular.
2. [tex]y=\frac{5}{4}x+6[/tex]
Step-by-step explanation:
The first equation of Exercise 1 is incomplete. Let's assume that it is:
[tex]2x + 3y =n[/tex]
Where "n" is a number.
First, it is important to remember that the equation of a line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope and "b" is the y-intercept.
By definition, the slopes of perpendicular lines are negative reciprocals.
1 . If you solve for "y" from the first equation, you get:
[tex]2x + 3y =n\\\\3y=-2x+n\\\\y=-\frac{2}{3}x+\frac{n}{3}[/tex]
You can identify that the slope is:
[tex]m=-\frac{2}{3}[/tex]
The second equation of the line is:
[tex]y=\frac{3}{2}x+4[/tex]
And its slope is:
[tex]m=\frac{3}{2}[/tex]
Since the slopes are negative reciprocals, the lines are perpendicular.
2. Given the first equation of the line:
[tex]y= -\frac{4}{5}x+6[/tex]
You can identify that:
[tex]m=-\frac{4}{5}\\\\b=6[/tex]
Since the first line and the second one are perpendicular, you know that the slope of the other line is:
[tex]m=\frac{5}{4}[/tex]
According to the information given in the exercise, both lines have the same y-intercept; therefore, the equation of the second line is:
[tex]y=\frac{5}{4}x+6[/tex]
