Question

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

Answer 1

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

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 =
  `(-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)` =
`(-4)/(-5)` =
`(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)` =
`(-2)/(3)`

∴ m
`_(2)` =
`-(2)/(3)`

→ By using the 2nd rule above

∵ tan Ф
`_(2)` =
`-(2)/(3)`

∴ Ф
`_(2)` =
`tan^(-1)`(
`-(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
`(m_(2)-m_(1))/(1+m_(1).m_(2))`I