Lines.Are the lines with equations 2x + y = 2 and x - 2y = 0 parallel, perpendicular or neither?
You need to convert the given forms of equations into slope intercept form `y = mx + b` (m - slope, b - y intercept).
Converting the original form `2x + y = ` 2 into slope intercept form, yields:
`y = -2x + 2 => m_(d_1) = -2`
Converting the original form `x - 2y = 0` into slope intercept form, yields:
`x - 2y = 0 => 2y = x => y = (1/2)*x => m_(d_2) = 1/2`
You may notice that the slopes have no equal values, hence, you need to test if the lines are perpendicular.
You need to use the equation that relates the slopes of two perpendicular lines, such that:
`m_(d_1) = -1/(m_(d_2)) => -2 = -1/(1/2) => -2 = -2`
Hence, testing if the lines are perpendicular yields that `d_1 _|_ d_2.`
If the lines are perpendicular, then the product of their slopes is -1.
For the beginning, we'll verify if the lines are intercepting. For this rason, we'll solve the system
2x + y = 2 (1)
x - 2y = 0 => x = 2y (2)
We'll substitute (2) in (1):
4y + y = 2
5y = 2
y = 2/5
x = 2*y
x = 4/5
The intercepting point is (4/5 ; 2/5).
Now, we'll write the equation in the standard form:
y = mx + n, where m is the slope and n is the y intercept.
2x + y = 2
We'll isolate y to the left side:
y = -2x + 2
The slope m1 = -2.
We'll put the 2nd equation in the standard form:
x = 2y
y = x/2
m2 = 1/2
Now, we'll verify if the product of the slopes gives -1:
m1*m2 = -1
-2*1/2 = -1
-1 = -1
Since the product is -1, the lines are perpendicular.