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

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

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step-by-step explanation:

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[tex]_{2}[/tex] - Ф[tex]_{1}[/tex]) 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 = [tex]\frac{-a}{b}[/tex]

∵ 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[tex]_{1}[/tex] = [tex]\frac{-4}{-5}[/tex] = [tex]\frac{4}{5}[/tex]

∴ m[tex]_{1}[/tex] = 0.8

→ By using the 2nd rule above

∵ tan Ф[tex]_{1}[/tex] = 0.8

∴ Ф[tex]_{1}[/tex] = [tex]tan^{-1}[/tex](0.8)

∴ Ф[tex]_{1}[/tex] = 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[tex]_{2}[/tex] = [tex]\frac{-2}{3}[/tex]

∴ m[tex]_{2}[/tex] = [tex]-\frac{2}{3}[/tex]

→ By using the 2nd rule above

∵ tan Ф[tex]_{2}[/tex] = [tex]-\frac{2}{3}[/tex]

∴ Ф[tex]_{2}[/tex] = [tex]tan^{-1}[/tex]( [tex]-\frac{2}{3}[/tex])

∴ Ф[tex]_{2}[/tex] = -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[tex]\frac{m_{2}-m_{1}}{1+m_{1}.m_{2}}[/tex]I

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