# Lines.Are the lines with equations 2x + y = 2 and x - 2y = 0 parallel, perpendicular or neither?

Asked on by happybee

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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.`

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

We’ve answered 319,627 questions. We can answer yours, too.