find angle between two lines

Question

Find the angle between two lines

x-2y=7

6x+2y=5

Answer 1

x-2y=7

6x+2y=5

Write the line equation in slope-intercept form y = mx + b, m is slope and b is y-intercept.

x-2y=7

x -2y -x = 7 -x                                        (Add each side by -x)

-2y = 7 -x

-2y/-2 = 7 -x/-2                                       (Divide  each side by -2)                                  

 y =  7 -x/-2                                            (Simplify) 

 y = (7/-2) - (x/-2 )       
                                                    

Compared to the slope intercept equation y = mx +b

slope(m₁) = 1/2, and intercept(b₁) = -7/2

And again the next equation Write the line equation in slope-intercept form y = mx + b, m is slope and b is y-intercept.

6x+2y=5

6x+2y -6x = 5 -6x                                   (Subtract each side by 6x)

2y = 5 -6x   

y = -(6/2)x +5/2                                       (Simplify)

y = -(3)x +5/2                                          (Simplify)

Compared to the slope intercept equation y = mx +b

slope(m₂) = -(3), and intercept(b₂) = 5/2

The Angle between two lines 

tan(θ) = (m₁ -m₂)/(1+m₁m₂)

Substitute the value of m₁ & m₂ above equation

tan(θ) = (1/2 - (-3))/(1+1/2(-3))

tan(θ) = (2)/(1 - (3/2))                               (LCM of 1/2 +3= 4/2=2)                    

tan(θ) =(2)/(- (1/2))                                   (LCM of 1 -(3/2) = - 1/2)

tan(θ) = - 4                                             (Simplify)

The value of (θ) = tan^-1 (-4) = -75.96.

Comments

tan(θ) = | (m₁ -m₂)/(1+m₁m₂) |.

tan(θ) = | (1/2 - (-3))/(1+1/2(-3)) |

tan(θ) = | [(1 + 6)/2] / [(1 - 3/2)] |

tan(θ) = | [7/2] / [(2 - 3)/2] |

tan(θ) = | [7/2] / [- 1/2] |

tan(θ) = | - 7 |

tan(θ) = 7.

The value of (θ) = tan^-1 (7) = 81.87^O.

Therefore, the angle between lines (θ) is 81.87^O .