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

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2x + y = 2

x - 2y = 0

First we will rewrite the equation in the slope format:

==> 2x + y = 2

==> y= -2x + 2

Then, the slope m1 = -2

Now we will write equation (2):

x - 2y = 0

Add 2y to both sides.

==> 2y = x

Now we will divide by 2.

==> y= (1/2) x

==> Then, the slope m2 = (1/2)

Now since m1 and m2 are different, then the lines are NOT parallel.

If the lines are perpendicular, then we know that m1*m2 = -1

==> -2 * 1/2 = -1

**Then, the lines are perpendicular.**

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.**