I have a linear function math problem, please walk me through the steps.?

Question

I missed this question on my most recent math test, but thought that I had worked it out right. Sorry that it's a long word problem but I am frustrated because I have worked through it so many times and am still coming out wrong.

The chart shows a country’s annual egg production. Model the data in the chart with a linear function, using the points (2, 57.7) and (4, 61.3) Predict egg production in 2008.

Question One: What is the linear model for the data?

Y=?

Question Two: What will egg production (in billions) be in 2008?

Here is the chart:

Year, x Egg Production (in billions), y

1994, 0 55.4

1995, 1 56.2

1996, 2 57.7

1997, 3 59.2

1998, 4 61.3

1999, 5 63.8

200, 6 65.7

I am not looking for just the answer but someone who can walk me through the steps and show me where I am going wrong. Thank you so much in advance.

Answer 1

alright so since they gave you those 2 points you first have to find the slope. (y2-y1)/(x2-x1) so it's gonna be (61.3-57.7)/(4-2). that simplifies to 9/5. that is your slope. so you have to substitute this into the y=mx+b format. you can pick either point they want you to use and that is your x and y. i'm going to use (4, 61.3) for now. now you solve for b. 61.3=(9/5)*4+b. b= 54.1. since you had to subtract the (9/5)*4. now your equation looks like y=(9/5)x+54.1. for the second part...since 2008 is 8 years off, you add 8 to 6 and then that is your new x=14. put that in the new equation you have and y=79.3. hope this helped.

Answer 2

Here's the part to focus on:

Model the data in the chart with a linear function, using the points (2, 57.7) and (4, 61.3)  Predict egg production in 2008.

First, it should be linear (a straight line) so find the slope:

change in y over change in x = 3.6 / 2 = 1.8

That represent the average annual change in production.

From there you have several ways to come up with the equation.  You can choose one of the two points and write an equation in point-slope form, or you could plug in one point and the slope into slope-intercept form (y = mx+b) and solve for the y-intercept (b).  I will use point-slope form:

y - y1 = m(x - x1)

y - 57.7 = 1.8(x - 2)

y - 57.7 = 1.8x - 3.6

y = 1.8x + 54.1

That is a linear model for this set of data, using the two points stated in the question.

Second, find the production in 2008.  That's year 14 (2008 - 1994), so plug 14 in for x, so you can calculate y.

y = 1.8x + 54.1

y = 1.8(14) + 54.1

y = 79.3 billion eggs in 2008

There ya go.

Just as a tip, one good way to verify the reasonableness of your answer is to graph all points, sketch a line of best fit through them, then look at the target year (in this case 2008) to see if it appears to be close to the line of prediction.  You can look up "linear regression" in a search engine for more info about the actual statistical procedures (it's a pain, I'd use a graphing calculator myself!)

Answer 3

Taking the two data points as 57.7 billion eggs in 1996 and 61.3 billion eggs in 1998, and assuming a linear relationship of eggs with years, it can be seen that the egg production increases at the rate of

(61.3 billion - 57.7 billion)/2 = 1.8 billion eggs per year. Therefore, for any future year yyyy, the linearization gives

eggs = 57.7 billion + 1.8 billion(yyyy - 1996) or

eggs = 61.3 billion + 1.8 billion(yyyy - 1998). Either of these equations is a linear model for the data.  Plugging year 2008 in the second equation, gives 61.3 billion + 10(1.8 billion) = 79.3 billion eggs