# 2x+8y=20 and 3x+9y=50 Parallel-Perpendicular-Intersecting

<-- Enter Line 1 Equation
<-- Enter Line 2 Equation (only if you are not pressing Slope)

We want to determine if 2x+8y=20 and 3x+9y=50 are parallel, intersect, or intersect and are perpendicular:

Simplify the equation for line 1 to get it into our y = mx + b format:
2x + 8y = 20
8y = -2x + 20

Divide each side of the equation by 8 to isolate y:
 8y 8
 =
 -2x + 20 8

Simplifying and evaluating, we have:
y = -0.25x + 2.5
Therefore, the slope (gradient) of line equation 1 = -0.25

Simplify the equation for line 2 to get it into our y = mx + b format:
3x + 9y = 50
9y = -3x + 50

Divide each side of the equation by 9 to isolate y:
 9y 9
 =
 -3x + 50 9

Simplifying and evaluating, we have:
y = -0.33x + 5.6
Therefore, the slope (gradient) of line equation 2 = -0.33

Since y = -0.25x + 2.5 and y = -0.33x + 5.6, set each line equation equal to each other and solve for x:
-0.25x + 2.5 = -0.33x + 5.6
-0.25x - -0.33x = 5.6 - 2.5
+ 0.333x = 3.6
x = 3.6/ + 0.333
x = 36.

Now that we have x, plug it into equation 1 to find y
y = -0.25 * (36.) + 2.5
y = -9.7 + 2.5
y = -6.6667
Therefore, our intersection point = (36., -6.6667)

Calculate the product of the 2 slopes:
Slope 1 * Slope 2 = -0.25 * -0.33 = 0.333
Since the product of the 2 slopes <> - 1, the lines are not perpendicular
The 2 lines intersect at (36., -6.6667)

Check to see if the system of equations is Independent:
Since the slopes are different and the lines cross, the systems are independent

Check to see if the system of equations is Dependent:
In order for a system to be dependent, the slopes and y-intercept must be the same. This is not the case

Check to see if the system of equations is Inconsistent:
In order for a system to be inconsistent, the slopes must be the same and y-intercept must different. This is not the case

Calculate the angle θ formed by the two lines:
 tan(θ) = m2 - m1 1 + m2m1

 tan(θ) = -0.33 --0.25 1 + -0.33 *-0.25

 tan(θ) = -0.333 1 + 0.333

 tan(θ) = -0.333 3

tan(θ) = -0.077
θ = -4.3987