Parallel Line to y = -4x + 3 through Point (0, -2): find slope, y-intercept

[Gadsilla]

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.

. . . . .$\displaystyle y\, =\, -4x\, +\, 3\qquad (0,\, -2)$

. . . . .$\displaystyle y\, =\, mx\, +\, b$

. . . . .$\displaystyle -2\, -\, \left(\dfrac{1}{4}\right)\, (0)\, =\, -2$

. . . . .$\displaystyle y\, =\, -\dfrac{1}{4}\, x\, -\, 2$

 

[mmm4444bot]

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.

----------------

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.
----------------

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.
----------------

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.
----------------

Given y = 2x - 9, find a parallel line …

Immediately write y = 2x + b

Same slopes.
----------------

Given y = (1/4)x - 5/4, find a perpendicular line …

Immediately write y = -4x + b

Negative reciprocal slope.
----------------

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

----------------

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:

 

[mmm4444bot]

PS: If you prefer, it's not necessary to post an image of basic exercises; you may simply type them out:

Find the slope and y-intercept of a line through (0,-2) which is parallel to the line y = -4x + 3

 

[Gadsilla]

I don't see where I went wrong.

 

[Harry_the_cat]

I don't see where I went wrong.

What did mmm444bot just tell you about the gradient of parallel lines?

The gradient of the given line is -4, therefore the gradient of any parallel line is also -4.

Why is your gradient 1/4 (or -1/4)? It is unclear what you have written.

 

[Gadsilla]

What did mmm444bot just tell you about the gradient of parallel lines?

The gradient of the given line is -4, therefore the gradient of any parallel line is also -4.

Why is your gradient 1/4 (or -1/4)? It is unclear what you have written.

I just realized I've been assuming it was a perpendicular line when it's a parallel line since all the previous questions it has been generating has been about perpendicular lines.

 

[Dr.Peterson]

I just realized I've been assuming it was a perpendicular line when it's a parallel line since all the previous questions it has been generating has been about perpendicular lines.

This could be an added benefit of typing out a problem rather than just attaching an image: in typing it out, the word "parallel" passing through your brain might trigger a realization of what it is saying, and you might not even need to post the question! (The main reason is to make it easier to search for a problem, and easier for helpers to copy and paste.)

Incidentally, this is very common in face-to-face tutoring, too: I ask the student to read the problem to me and explain their thinking, and in talking about it they will solve their own problem.

 

[Gadsilla]

So

y= -4x+b
-2-(-4)(0)=-2
y=-4x+(-2)

 

[Dr.Peterson]

So

y= -4x+b
-2-(-4)(0)=-2
y=-4x+(-2)

Correct. We can also write it as y = -4x - 2.

You can check by seeing that the point (0,-2) is on the line: -4(0) - 2 = -2.

You may also notice that (0.-2) is on the y-axis, so it is the y-intercept, and you could have used the slope-intercept form directly, knowing already that b = -2.

 

[Gadsilla]

Correct. We can also write it as y = -4x - 2.

You can check by seeing that the point (0,-2) is on the line: -4(0) - 2 = -2.

You may also notice that (0.-2) is on the y-axis, so it is the y-intercept, and you could have used the slope-intercept form directly, knowing already that b = -2.

What does "slope-intercept form" mean ? It's where I got stuck on the last question regarding the perpendicular line as well, finding the slope.

 

[Deleted member 4993]

What does "slope-intercept form" mean ? It's where I got stuck on the last question regarding the perpendicular line as well, finding the slope.

I would first ask "google" with keywords - slope, intercept, line. See what do you find - and tell us.

 

[pka]

View attachment 10510
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.

Once again I advocate for the standard form of the line.
Any line that is parallel to $\displaystyle ax+by+c=0$ looks like $\displaystyle ax+by+d=0$
So the given line in standard form is $\displaystyle 4x+y-3=0$ so the parallel line is $\displaystyle 4x+y+d=0.$
What is $\displaystyle d~?$ Well use the given point: $\displaystyle 4(0)+(-2)+d=0$.

 

[Dr.Peterson]

What does "slope-intercept form" mean ? It's where I got stuck on the last question regarding the perpendicular line as well, finding the slope.

Have you noticed that this site turns certain words into links to articles on the site? The link for the word "intercept" (in my quote from you above, for example) takes you to an explanation of slope-intercept form. Take a look, and then ask any questions you have about it.

 

[Gadsilla]

What does "slope-intercept form" mean ? It's where I got stuck on the last question regarding the perpendicular line as well, finding the slope.

Have you noticed that this site turns certain words into links to articles on the site? The link for the word "intercept" (in my quote from you above, for example) takes you to an explanation of slope-intercept form. Take a look, and then ask any questions you have about it.

I have noticed, and clicked it. I also googled it, but it doesn't make sense to me.

 

[Dr.Peterson]

I have noticed, and clicked it. I also googled it, but it doesn't make sense to me.

So now you do what I suggested, and ask specific questions about what you've read. We can't know what parts don't make sense to you without knowing what you are thinking. Tell us what you think it is saying, and where you are confused, so we can give the specific answers you need.

 

[Gadsilla]

So now you do what I suggested, and ask specific questions about what you've read. We can't know what parts don't make sense to you without knowing what you are thinking. Tell us what you think it is saying, and where you are confused, so we can give the specific answers you need.

It's hard to explain because none of it makes sense. I'm not following at all. I am at the same spot in regards to knowledge I was when I first asked the questions.

I don't know how to proceed in finding the slope.

 

[Deleted member 4993]

It's hard to explain because none of it makes sense. I'm not following at all. I am at the same spot in regards to knowledge I was when I first asked the questions.

I don't know how to proceed in finding the slope.

Did you find a definition of the "slope of the line"?

If you did - what was it (whether it made any sense to you or not)?

Just tell us what you found as the definition.

We can go word-by-word and expound on that!

 

[Gadsilla]

Did you find a definition of the "slope of the line"?

If you did - what was it (whether it made any sense to you or not)?

Just tell us what you found as the definition.

We can go word-by-word and expound on that!

Yes, I found how to do it. However, the explanation isn't really telling me anything.

"To calculate the slope of a line you need only two points from that line, (x1, y1) and (x2, y2). The equation used to calculate the slope from two points is: On a graph, this can be represented as: There are three steps in calculating the slope of a straight line when you are not given its equation."

 

[mmm4444bot]

Once again I advocate for the standard form of the line.

I agree there's a time and place for that, but Gadsilla isn't there, yet.

My opinion: When a student receives an exercise in one form, that's generally the form we ought to use.

I hope everyone is also mindful that Gadsilla doesn't yet have a good foundation in arithmetic or pre-algebra topics (eg: unable to subtract an improper fraction from a Whole number).

 

[mmm4444bot]

Gadsilla, I would like some background information. If you're willing, please answer the following. The answers will help us provide you with appropriate guidance/suggestions. :cool:

Are you a returning student?

Are you currently enrolled in a math class?

Are you self-studying?

What is your math background?

How long has it been, since you previously studied math?

Why are you studying math?