Question

1. Determine whether the lines given by the equations 2x + 3y = and y=3/2x+4

Answer 1

Answer:

1. Yes, the lines are perpendicular.

2. $y=\frac{5}{4}x+6$

Step-by-step explanation:

The first equation of Exercise 1 is incomplete. Let's assume that it is:

$2x + 3y =n$

Where "n" is a number.

First, it is important to remember that the equation of a line in Slope-Intercept form is:

$y=mx+b$

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:

$2x + 3y =n\\\\3y=-2x+n\\\\y=-\frac{2}{3}x+\frac{n}{3}$

You can identify that the slope is:

$m=-\frac{2}{3}$

The second equation of the line is:

$y=\frac{3}{2}x+4$

And its slope is:

$m=\frac{3}{2}$

Since the slopes are negative reciprocals, the lines are perpendicular.

2. Given the first equation of the line:

$y= -\frac{4}{5}x+6$

You can identify that:

$m=-\frac{4}{5}\\\\b=6$

Since the first line and the second one are perpendicular, you know that the slope of the other line is:

$m=\frac{5}{4}$

According to the information given in the exercise, both lines have the same y-intercept; therefore, the equation of the second line is:

$y=\frac{5}{4}x+6$