Question

Determine the angle between the two lines with equations 4x - 5y = 11 and 2x + 3y = 7.

Answer 1

Answer:

The angle between the two lines is 72.35° ⇒ to the nearest hundredth

Step-by-step explanation:

• The angle between the two lines is the difference between their angles with the positive part of the x-axis (Ф = Ф$_{2}$ - Ф$_{1}$)

• The slope of the line equal to the tangent of the angle between the line and the positive part of the x-axis (m = tan Ф)

• The slope of the line whose equation is ax + by = c is m = $\frac{-a}{b}$

∵ The equation of the first line is 4x - 5y = 11

∴ a = 4 and b = -5

→ By using the 3rd rule of the slope above

∵ m$_{1}$ = $\frac{-4}{-5}$ = $\frac{4}{5}$

∴ m$_{1}$ = 0.8

→ By using the 2nd rule above

∵ tan Ф$_{1}$ = 0.8

∴ Ф$_{1}$ = $tan^{-1}$(0.8)

∴ Ф$_{1}$ = 38.66° ⇒ to the nearest hundredth

∵ The equation of the first line is 2x + 3y = 7

∴ a = 2 and b = 3

→ By using the 3rd rule of the slope above

∵ m$_{2}$ = $\frac{-2}{3}$

∴ m$_{2}$ = $-\frac{2}{3}$

→ By using the 2nd rule above

∵ tan Ф$_{2}$ =  $-\frac{2}{3}$

∴ Ф$_{2}$ = $tan^{-1}$( $-\frac{2}{3}$)

∴ Ф$_{2}$ = -33.69° ⇒ to the nearest hundredth

→ By using the 1st rule above

∵ Ф = -33.69 - 38.66

∴ Ф = -72.35°

→ Ignore the negative sign

∴ The angle between the two lines is 72.35° ⇒ to the nearest hundredth

V.I.N:

You can use this rule to find the angle between 2 lines tan Ф = I$\frac{m_{2}-m_{1}}{1+m_{1}.m_{2}}$I