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Of greater importance, notice that this exercise nowhere said anything about parallel or perpendicular lines, nor directed us to find any line's equation. For instance, you would simply not be able to tell, just "by looking" at the picture, that drawn lines with slopes of, say, m 1 = 1. I start by converting the "9" to fractional form by putting it over "1". So: The first thing I'll do is solve "2x − 3y = 9" for " y=", so that I can find my reference slope: So the reference slope from the reference line is. There is one other consideration for straight-line equations: finding parallel and perpendicular lines.
It turns out to be, if you do the math. ] The distance turns out to be, or about 3. Now I need to find two new slopes, and use them with the point they've given me; namely, with the point (4, −1). Then click the button to compare your answer to Mathway's. Then I can find where the perpendicular line and the second line intersect. 00 does not equal 0. The slope values are also not negative reciprocals, so the lines are not perpendicular. Since a parallel line has an identical slope, then the parallel line through (4, −1) will have slope. This negative reciprocal of the first slope matches the value of the second slope. Equations of parallel and perpendicular lines. Therefore, there is indeed some distance between these two lines. Parallel lines and their slopes are easy.
To answer the question, you'll have to calculate the slopes and compare them. Note that the distance between the lines is not the same as the vertical or horizontal distance between the lines, so you can not use the x - or y -intercepts as a proxy for distance. In other words, to answer this sort of exercise, always find the numerical slopes; don't try to get away with just drawing some pretty pictures. Nearly all exercises for finding equations of parallel and perpendicular lines will be similar to, or exactly like, the one above. Perpendicular lines are a bit more complicated. I'll find the values of the slopes. Then my perpendicular slope will be. For the perpendicular slope, I'll flip the reference slope and change the sign. Here is a common format for exercises on this topic: They've given me a reference line, namely, 2x − 3y = 9; this is the line to whose slope I'll be making reference later in my work.
The first thing I need to do is find the slope of the reference line. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will have to be a decreasing line). Here's how that works: To answer this question, I'll find the two slopes. Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel. And they then want me to find the line through (4, −1) that is perpendicular to 2x − 3y = 9; that is, through the given point, they want me to find the line that has a slope which is the negative reciprocal of the slope of the reference line. Otherwise, they must meet at some point, at which point the distance between the lines would obviously be zero. ) Are these lines parallel? So perpendicular lines have slopes which have opposite signs. Content Continues Below. In your homework, you will probably be given some pairs of points, and be asked to state whether the lines through the pairs of points are "parallel, perpendicular, or neither".
With this point and my perpendicular slope, I can find the equation of the perpendicular line that'll give me the distance between the two original lines: Okay; now I have the equation of the perpendicular. These slope values are not the same, so the lines are not parallel. Then I flip and change the sign. This is just my personal preference. This is the non-obvious thing about the slopes of perpendicular lines. ) Pictures can only give you a rough idea of what is going on. That intersection point will be the second point that I'll need for the Distance Formula. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down.
I'll solve for " y=": Then the reference slope is m = 9. I can just read the value off the equation: m = −4. Note that the only change, in what follows, from the calculations that I just did above (for the parallel line) is that the slope is different, now being the slope of the perpendicular line. If I were to convert the "3" to fractional form by putting it over "1", then flip it and change its sign, I would get ". Then the full solution to this exercise is: parallel: perpendicular: Warning: If a question asks you whether two given lines are "parallel, perpendicular, or neither", you must answer that question by finding their slopes, not by drawing a picture! To finish, you'd have to plug this last x -value into the equation of the perpendicular line to find the corresponding y -value. The only way to be sure of your answer is to do the algebra. I'll solve each for " y=" to be sure:.. Then the answer is: these lines are neither. I know I can find the distance between two points; I plug the two points into the Distance Formula. To give a numerical example of "negative reciprocals", if the one line's slope is, then the perpendicular line's slope will be. Then the slope of any line perpendicular to the given line is: Besides, they're not asking if the lines look parallel or perpendicular; they're asking if the lines actually are parallel or perpendicular. 99 are NOT parallel — and they'll sure as heck look parallel on the picture.
Since these two lines have identical slopes, then: these lines are parallel. But even just trying them, rather than immediately throwing your hands up in defeat, will strengthen your skills — as well as winning you some major "brownie points" with your instructor. Remember that any integer can be turned into a fraction by putting it over 1. And they have different y -intercepts, so they're not the same line. This would give you your second point. 99, the lines can not possibly be parallel. Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade. So I'll use the point-slope form to find the line: This is the parallel line that they'd asked for, and it's in the slope-intercept form that they'd specified. For the perpendicular line, I have to find the perpendicular slope. It's up to me to notice the connection. Where does this line cross the second of the given lines? I could use the method of twice plugging x -values into the reference line, finding the corresponding y -values, and then plugging the two points I'd found into the slope formula, but I'd rather just solve for " y=". This line has some slope value (though not a value of "2", of course, because this line equation isn't solved for " y=").
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