Linear equation word problem

Question

Tracy Finds that there is a linear relationship between the number of hours she studies for a test and the grade she earns on thee test . If she studies for 3 hours for her math test she receives a 65, however if she studies for 6 hours she receives a 95. Write the equation of a line that this situation describes. What would you expect her score to be if she did not study at all

Answer 1

This problem looks like you are given two points on the line (linear relationship).  For 3 hours study Tracy gets a grade of 65 and for 6 hours she gets 95.  In algebra, those points – (s,g) --    are (3, 65) and (6,95).

 

So,

     The slope is (95-65)/ (6-3) = 30/3 = 10.

     In slope-intercept form, the equation is     g = 10s + b

            Plugging in either point gives  b = 35, the g-intercept (value of g when s=0, not study at all).

 

Now, unless Tracy can get a grade of more than 100, there must be allowable values for s (study time).  Can you determine the maximum value for s if the maximum grade is 100.    (these are called domain and range)

Answer 2

Let's start by taking a look at the slope. We can solve with the formula:

(y₂ - y₁)/(x₂ - x₁)

 

Which is the x and which is the y? We look at which value is dependent on the other. Since grades are based on the hours studied, then x is the independent value and y the dependent, giving us (3, 65) and (6, 95).

slope = (95 - 65)/(6 - 3)

slope = 30/3

slope = 10

 

We can use the point slope formula and transform that into slope-intercept form. I'll pick the point (3, 65), but either one will work:

y - y₁ = m(x - x₁)

y - 65 = 10(x - 3)

y - 65 = 10x - 30

y = 10x + 35

 

You can check the answer by plugging in either 3 or 6 and see if they come out to 65 or 95 respectively.

0 hours studying would be our y-intercept, or the value of b. Here, it is 35, making that her score after 0 hours of study.

Answer 3

Grade= 10* hours+ 35

You would expect her score to be 35 if she did not study at all.

Since her grade improves 30 points in 3 hours it therefore increases 10 points per hour. Taking away 30 points for the 3 hours studied from the 65 she would get means that with no studying she would receive a 35, 65-30,  with no time studying.