Problem Solving: Linear Equation

Question

1. Problem Solving: Linear Equation

Supposed you have given a positive five series different intercept or constant value starting from its point of origin. Each coefficient are twice the value of each constant and each x variable are trice the value of its coefficient.

1.1  Solve the values of y, plot your solution here:

1.2. Upload the Graphical Representation of the problem above.

Comments

Please clarify your problem.

1. Do you mean 5 linear equations or a polynomial with degree 5?  (The heading "linear equation" would seem to rule out polynomials.) Please clarify "positive five series".
2. Is it y-intercept(s) or x-intercept(s)?
3. By origin do you mean the graph(s) of the function pass through or start at (0,0)?
4. By trice do you mean thrice or twice?
5. A coefficient is a constant, so how can a multiple of a constant be the variable x?

Your wording suggests a paraphrase of the original question, because the English does not make grammatical sense. It would be helpful if you could provide a photo (screenshot) of the original question, which can be uploaded to this website using the menu feature from the menu bar.

Answer 1

Hey Rod, I'm not really good at any Math's whatsoever so your guess is as good as mine.

1.) I think it's 5 linear equations.

2.) I think it's y-intercepts.

3.) I think yes.

4.) I think thrice.

5.) I have no clue, so, be my guess to answer the question.

Comments

Many thanks, pfragment, for your timely suggestions. Let's see what Karl says now.

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Rod, pfragment and Karl is the same user. I just created an account named pfragment because I cannot used reply without an account.

-Karl

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Thank you for your prompt response. Can you give me an example of any recent question you may have had from the same or similar source, so that I may be able to relate your problem through the context of what you are currently studying? There are so many errors in the English presentation of this problem and I'm finding it difficult to understand what the question is getting at! The "positive five series" is really puzzling, as I can't relate it to anything meaningful. A linear equation which starts or passes through the origin is of the form y=mx where m is the gradient. There's no constant and the intercept is zero. So five linear equations would each have a zero y-intercept. That's puzzling in view of what the question is asking for. So I'm looking for an interpretation that fits what seems to be the facts.

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The only thing I can give example of is where our instructor source his lecture. This is the YouTube video where he gets his example. It's very long, sorry. This is Quantitative Methods.

Video 1: Introduction to Simple Linear Regression

Video 2: Confidence Intervals

It's okay Rod if you don't answer the question. Don't stress yourself.

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Thanks again for being so quick! I'll check out the videos, which seem to suggest statistics, so I'm beginning to think we're talking about linear regression, which is quite a different thing to linear equations. A question worth 50 points is more likely to be a statistical one rather than simple algebra. I'll let you know by comment how I get on. Don't worry: I won't get too stressed over this one! The videos might clarify the question.

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The videos are very good in explaining how linear equations and linear regression are related. The question you were given could be related to the early part of Video 1. There doesn't seem to be a relation between your problem and scatter diagrams and data sampling, and certainly doesn't involve confidence intervals. My guess is that the problem you were given is to draw several straight lines (graphs of linear equations) having either the same y-intercept or having the same x-intercept, but different slopes. The x-intercept may be what the question calls "point of origin". The x-coefficient is the slope while the y-intercept is the constant. Since you were given no sample data, I think the problem is either:

(1) to draw five different arbitrary lines starting at the same x-intercept (some negative x value) and having different slopes and therefore different y-intercepts; or

(2) to draw five different arbitrary lines starting at different x-intercepts but passing through the same y-intercept and having different slopes. 

As for the twice and thrice part, well, I'm not sure exactly how to interpret these geometrically on the graphs. To me, it suggests a slope of 2/3 for every line---a rise of 2 units for every 3 units horizontally. So all the lines would have the same slope but different x- and y-intercepts. Then you would write the equations relating the dependent variable y to the independent variable x.

I think it would be a good idea for you to ask your tutor to explain what the problem is asking for in view of the poor English and lack of clarity, and how it relates to the videos. Or you could discuss it with fellow students. (Then show your tutor what you think was meant.)

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See my revised answer.

Answer 2

"Each coefficient [is] twice the value of its constant" I interpret as:

y=2a_ix+a_i, where a_i is the constant of the i-th equation (i is between 1 and 5). a_i is also the y-intercept.

Or perhaps "Each [y-]coefficient [is] twice the value of its constant and each x variable [is] t[h]rice the value of its [y-]coefficient" meaning:

2a_iy-6a_ix-a_i=0, which is the same as:

2a_iy=6a_ix+a_i. (If the y-coefficient is 2a_i, then thrice this coefficient is 6a_i.)

Neither of these interpretations leads to linear equations starting at or passing through the origin (0,0), so we're looking at what "point of origin" could mean. It could mean that all the linear equations intersect at one point which is taken to be the point of origin. But see below.

1.1 y=(6a_ix+a_i)/(2a_i)=3x+½ so there is only one actual equation, because the coefficients cancel out. So this is just a straight line.

More attempts to interpret to follow...

There's no more information in the question so I assume the value of each a_i is arbitrary.

Below is my best interpretation of this problem.

Table of values used to draw the graph:

β0	1	2	3	4	5
β1	2	4	6	8	10
data value x	3	6	9	12	15
y=	1+2x	2+4x	3+6x	4+8x	5+10x
Point on line	F(3,7) green	G(6,26) blue	H(9,57) red	I(12,100) orange	J(15,155) purple

Points A(0,1), B(0,2), C(0,3), D(0,4), E(0,5) are the intercepts or constants.

Note that all lines intersect at (-0.5,0).

 

Comments

Thanks a lot Rod for the answer. It really helps.