find angle between two lines

Question

Find the angle between two lines

2x-y=2

4x+3y=24

Answer 1

Write the line equation in slope-intercept form y = mx + b, m is slope and b is y-intercept. 2x-y=2 2x -y +y = 8 +y                                  (Add y to each side) 2x = 8 +y y +8 -8 = 2x -8                                   (Subtract 8 to each side) y = 2x -8                                           (Subtract: 8 -8 = 0) Compared to the slope intercept equation y = mx +b slope(m1) = 2, and intercept(b1) = -8 And again the next equation Write the line equation in slope-intercept form y = mx + b, m is slope and b is y-intercept. 4x+3y=24 4x+3y -4x = 24 -4x                            (Subtract 4x to each side) 3y = 24 -4x    y = -(4/3)x +24/3                                (Simplify) y = -(4/3)x +8                                    (Simplify) Compared to the slope intercept equation y = mx +b slope(m2) = -(4/3), and intercept(b2) = 8 The Angle between two lines  tan(θ) = (m1 -m2)/(1+m1m2) Substitute the value of m1 & m2 above equation tan(θ) = (2 - (4/3))/(1+2(-(4/3)))              = (2/3)/(1 - (8/3))                    (LCM of 2 -4/3= 2/3)                                  =(2/3)/(- (5/3))                        (LCM of 1 -(8/3) = - 5/4)              = - 2/5                                  (Simplify) The value of (θ) = tan-1 (-2/5)  = -21.80o

 

Comments

tan(θ) = | (m₁ -m₂)/(1+m₁m₂) |.

Here, m₁ =2 and m₂ = - 4/3

tan(θ) = | (2 -(-4/3))/(1+(2)(-4/3)) |.

tan(θ) = | (2 +(4/3))/(1-(8/3))|.

           = | (10/3)/(-5/3)|.

           = |-2|.

           =2

θ = tan^-1 (2) = 63.43^O.

Therefore, the angle between lines (θ) is 63.43^O .

 

Answer 2

Given two lines are

2x-y =2

4x+3y = 24

Let convert  first line eqation is in slope intercept form

2x-y = 2

-y = -2x+2

Multiply negitive sign in each sides eqation

y = 2x-2

Slope of the line say m1= 2

Let convert the second line eqation is in slope intercept form

4x+3y = 24

Subtract 4x each sides

4x-4x+3y = 24-4x

3y = -4x+24

Apply division property each sides divide by 3

3y/3= -4/3x+24/3

y = -4/3x+8

Slope of the second line say m2 = -4/3

Angle between  two lines Tan theta = (+/-)(m1-m2)/1+(m1*m2)

=(+/-)[2-(-4/3)]/1+[2*(-4/3)]

=(+/-)[2+4/3]/1-8/3

=(+/-)(10/3)/(-5/3)

=(+/-)10/5

=(+/-)2

Angle between two lines 2x-y = 2 , 4x+3y = 24

Tan theta = 2

Comments

tan(θ) = | (m₁ -m₂)/(1+m₁m₂) |.

θ = tan^-1 (2) = 63.43^O.

Therefore, the angle between lines (θ) is 63.43^O .