An equation of a line through (0, -2) which is parallel to the line y = -4x + 3 has what slope and y-intercept?
So it started to make sense, and I switched it up with another question and it completely throws me off. I'm at a random question generator I've got for practice.
. . . . .\(\displaystyle y\, =\, -4x\, +\, 3\qquad (0,\, -2)\)
. . . . .\(\displaystyle y\, =\, mx\, +\, b\)
. . . . .\(\displaystyle -2\, -\, \left(\dfrac{1}{4}\right)\, (0)\, =\, -2\)
. . . . .\(\displaystyle y\, =\, -\dfrac{1}{4}\, x\, -\, 2\)
Please familiarize yourself with the forum's
guidelines. We need to start a new thread for each new exercise. Also, I moved your threads off of the Calculus board because these exercises are introductory algebra.
I can't read your image; it's not enlarging for me, for some reason.
You're working on a basic theme; they give you the equation of a known line and then ask you to find a new line passing through a given point. The new line is specified to be either PARALLEL to the given line or PERPENDICULAR to it. Those words in caps relate to the slope, and that's what you need to focus on first.
Memorize this: parallel lines have the SAME slope, and perpendicular lines have slopes that are NEGATIVE RECIPROCALS.
This allows you to immediately write the form y=mx+b for the requested line, replacing m with the appropriate slope.
EGs
Given y = 7x - 3, find a parallel line passing through …
Immediately write: y = 7x + b
Parallel lines have the same slope, so m must be the same as the given line.
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Given y = (3/8)x + 1/8, find a parallel line through …
Immediately write: y = (3/8)x + b
Parallel lines have the same slope, so m is 3/8 in the new line, too.
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Given y = (-11/13)x + 9/13, find a perpendicular line …
Immediately write: y = (13/11)x + b
Perpendicular line has a negative reciprocal, for the slope.
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Given y = 16x - 11, find a perpendicular line …
Immediately write y = (-1/16)x + b
New m is the negative reciprocal, because they want a perpendicular line.
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Given y = 2x - 9, find a parallel line …
Immediately write y = 2x + b
Same slopes.
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Given y = (1/4)x - 5/4, find a perpendicular line …
Immediately write y = -4x + b
Negative reciprocal slope.
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See how this works? You have a given slope, and you're asked about a new line that's either parallel or perpendicular. So, right away, you can write the new line:
y = mx + b by setting slope m to the appropriate value (same as given slope for parallel OR the negative reciprocal for perpendicular).
Now we find b (the y-intercept). Here's the first example, again, with the coordinates of a given point:
Given y = 7x - 3, find a parallel line which passes through the point (1,1)
The new line is y = 7x + b, right? (Parallel line, same slope)
To find b, we now substitute the given coordinates for the variable symbols x and y:
y = 7x + b
1 = 7(1) + b
1 = 7 + b
Solve for b, by subtracting 7 from each side
-6 = b
The new line's equation is y = 7x - 6
The new line has slope 7 and y-intercept -6
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Given 2x - 9, find a parallel line passing through the point (-3,10)
Write: y = 2x + b
Substitute x=-3 and y=10
10 = 2(-3) + b
Do the multiplication: 10 = -6 + b
Solve for b, by adding 6 to each side: 16 = b
The new line is y = 2x + 16
The slope is 2, and the y-intercept is 16
Have I written anything that you're not sure about? If so, please ask. Otherwise, give your exercise another go. It looks like the slope is not correct, in your last line of work. :cool:
