find the value of the acute angle between the lines 3x-y+1=0

Question

x-2y+1=6

Answer 1

First, we need to find the slopes of the two lines by putting them in slope-intercept form (y = mx + c).

The first line:
3x - y + 1 = 0
-y = -3x - 1
y = 3x + 1

The slope of the first line is 3.

The second line:
x - 2y + 1 = 6
x - 2y = 5
-2y = -x + 5
y = 1/2x - 5/2

The slope of the second line is 1/2.

The acute angle between two lines is given by the formula:
tanθ = |(m1 - m2) / (1 + m1 * m2)|

where m1 and m2 are the slopes of the two lines.

Substitute the slopes into the formula:
tanθ = |(3 - 1/2) / (1 + 3 * 1/2)|
tanθ = |(5/2) / (5/2)|
tanθ = 1

Now, to find the angle θ, we take the arctan of 1:
θ = arctan(1)
θ = 45°

Therefore, the acute angle between the two lines is 45 degrees.