In 2005, the percent of U.S. drivers wearing seat belts was 82%. The number of drivers
wearing seat belts in 2000 was 71%. Let y be the number of drivers wearing seat belts in
the year x where x = 0 represents the year 2000.
a) Write a linear equation that models the percent of U.S. drivers wearing seat belts in terms
of the year x.
b) Use the equation found in part (a) to predict the percentage of U.S. drivers wearing seat
belts in the year 2009.
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It's like y = mx + c, where y is dependent variable when is linearly dependent upon independent variable x.
In the above problem, let's assume x denotes the year and y be the percentage of drivers wearing seat belt. We did this because we want to predict the percentage of drivers (that is dependent variable) from year ( which is independent variable).
Now all that's left is to find the value of m and c. Did you get it till now?
Ok good. Now all whats left is very simple.
We know that in the year 2000 we had 71% of US drivers wearing seat belt.
In mathematical terms when x = 0, we have y = 71.
Similarly when x = 5, we have y = 82. (Here I have assumed that x = 5 represents the year 2005, though its not explicitly mentioned in the problem but seeing the fact that x = 0 represents year 2000 I think its a reasonable assumption)
Now push these numbers into the equation y = mx + c one at time.
When x = 0, we have y = 71,
that means 71 = m*0 + c
which gives us the value of c = 71.
Did you get this?
a little confused now
u said that y=71, now youre saying c=71
Well, y is a variable. It's value depends upon x. When we are given a particular x we can find the value of y. On the other hand m and c are constant which means their value won't change no matter what the value of x and y is.
So, what we did was that we know that y = 71 when x = 0.
We substitute this value in the equation y = mx + c.
71 = m * 0 + c
71 = 0 + c
c = 71
Hence we find the value of c.
Does this make anything clear?
So here's the summary when x = 0 we had y = 71 and we also had c = 71. But notice that since c is a constant, so no matter what the value of x and y is, the c will remain 71 all the time.
okay, yes i got it. so for part A of my question, the equation would be, 71=m(0)+c?
There's one more thing left to do till we can give the answer to the first part.
We need to find the value of m. Once we find the value of m we can answer the first part.
I know it might not make sense to you but hold down for a bit.
We know that when x = 5, we had y = 82,
Now we substitute the value of x and y in the equation y = mx + c.
Notice that our purpose is to find the constants m and c. We know c, so just need to find m.
y = mx + c
From last time c = 71.
82 = m*5 + 71
5*m = 82 - 71
5*m = 11
m = 11/5
So, now we have m = 11/5 and c = 71.
Now we can answer the first part.
The answer is y = 11/5*x + 71
Notice that if you put x = 0 in our answer you will get y = 71 and if you put x = 5 you will get 82. That's the whole essence of linear equations. We have encoded the information provided to us in the question into this one line equation.
With out equation in hand we can compute the second part which is to find the percentage of US Drivers in the year 2009.
All you need to do now is to put x = 9 and watch the value of y,
Here we go,
y = 11/5*9 + 71 = 90.8
This is the answer for the second part.