Math CentralQuandaries & Queries


Question from Blake, a student:

I need to know what the formula for working a problem. I need to know how to find if i line is parallel or perpendicular by only looking at the equation. i know you must know the slope but how do i find slope?

Hi Blake.

Look in your textbook for the part where it talks about the equation y = mx + b. That's called the "slope-intercept" form of a line's equation. You can take any equation of a line and turn it into this form. The number in front of the "x" (including the minus sign if there is one) is the slope.

Here are some examples:

  1. 3x + y = 8
    y = -3x + 8
    so here m is -3. The slope is -3.

  2. x - 2y + 20 = 0
    -2y = -x - 20
    y = 0.5x + 10
    so here the slope is 0.5 (one half).

  3. x = 3y - 24
    3y = x + 24
    y = (1/3)x + 8
    so here the slope is 1/3 (one third, or about 0.333).

In each case, I re-arrange things so that y = something. Once it is in y = mx + b form, I can just "read off" the slope (the m value) and also the y-intercept (the b value).

Now let's say I am asked if two different equations are parallel, perpendicular or neither. Parallel lines have the same slope. Perpendicular lines have slopes that are the negated reciprocal of one another (for example, the negated reciprocal of 3 is -(1/3), also, the negated reciprocal of -(22/7) is 7/22).

  1. 3x + y = 8 and 2y + 6x = 1.
    I first turn each into y = mx + b (slope-intercept) form:
    3x + y = 8
    y = -3x + 8
    2y + 6x = 1
    2y = -6x + 1
    y = -3x + 0.5
    Notice that the m value of the first (y = -3x + 8) is -3. The m value of the second (y = -3x + 0.5) is also -3. The number and the sign have to match and here they do. So these two lines are parallel to each other.

  2. x - 2y + 20 = 0 and y = -2x + 1
    slope intercept form for the first is y = 0.5x + 10 (I did this one up above)
    the second equation is already in slope intercept form: y = -2x + 1
    So the slope of the first is 0.5 and the slope of the second is -2. Clearly, they aren't the same, so the lines are not parallel. But are they perpendicular? What is the negated reciprocal of -2? It is +(1/2) which is 0.5. So these two lines are indeed perpendicular.

  3. y = (1/3)x + 8 and y - 3x = 18.
    the first is already in slope intercept form. y = (1/3)x + 8.
    slope intercept form for the second is y = 3x + 18.
    So the first line has slope of (1/3) and the second slope is 3. Are these lines parallel or perpendicular?
    Answer is neither. Remember that we have to change the sign for perpendicular slopes.

Hope this helps,
Stephen La Rocque.

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