Question 1018759: mr santos drives from his home to his office in makati. as he gets nearer to his office, the distance from his office varies linearly on the number of minutes that he has been driving . he noticed that after 8 minutes of driving, he is already 13 kilometers away from his office. upon looking from the second time after 12 minutes, he is already 7 kilometers away
its about linear functions and their applications
thank you
Found 2 solutions by josmiceli, Theo:
Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website! I'm not sure what the question is, but I think
you want to know what the equation for this
trip is.
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Let  = the distance in km from his home to his office
When he first checks the time, his rate ( slope ) is:
-------------------
When he next checks the time, his rate ( slope ) is:
-------------------
Since the rate is linear, these slopes must be equal
Multiply both sides by 
 km
----------------
The slope is:
------------------
The form of the equation is:
 since it starts at ( 0,0 )
( y-intercept is  )
----------------
where is km and  in min
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check:
Equation must go through:
( 8, d-13 ) = ( 8, 25 - 13 ) =
( 8,12 )
OK
----------------
Equation must go through:
( 12, d - 7 ) = ( 12, 25 - 7 ) =
( 12, 18 )
OK
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! after 8 minutes, he's 13 kilometers away.
after 12 minutes, he's 7 kilometers away.
presumably they want to know how many minutes to get to his office, or possibly how many kilometers from his home to his office.
what you have here is two points from which you can calculate a slope and then from which you can calculate the y-intercept, and from which you can calculate the x-intercept.
we start with the points.
if we let x represent the time and y represent the distance, then the general equation is y = mx + b, where m is the slope and b is the y-intercept.
your points are (x1,y1) and (x2,y2)
(x1,y1) = (8,13).
(x2,y1) = (12,7)
m is the slope which is equal to (y2-y1)/(x2-x1).
this becomes (13-7) / (8-12) which is equal to 6 / -4 which is equal to -3/2.
now that you know the slope, you can solve for the y-intercept.
your equation becomes y = -3/2 * x + b
to find b, use either one of the points and replace x and y with their respective values.
we'll use the point (12,7).
y = -3/2 * x + b becomes 7 = -3/2 * 12 + b
solve for b to get b = 7 + (3/2 * 12) which is equal to 25.
your equation becomes y = -3/2 * x + 25.
the graph of your equation is shown below.
look below the graph for further comments.
when x = 8, y = 13
when x = 12, y = 8
when x = 0, y = 25
when x = 16 and 2/3, y = 0
when x = 8, y = 13, and when x = 12, y = 8, are the two points we used to create the line.
when x = 0, y = 25 is the y-intercept.
this is the value of y when x is equal to 0.
this tells you the total distance from home to office.
when y = 0, x = 16 and 2/3 is the x-intercept.
this is the value of x when y is equal to 0.
this tells you the total time it took to get from home to office.
the slope is telling you the rate that he is traveling at.
it's the change in the distance divided by the change in time.
the slope is -3/2.
this means that he is traveling 3/2 miles each minute.
his rate if 3/2 miles per minute.
his equivalent rate will be 3/2 * 60 miles per hour because there are 60 minutes in an hour.
that comes out to be 90 miles per hour.
he was moving pretty fast.
the basic formula that relates time and distance and speed is rate * time = distance.
he was 25 miles away.
it took him 16 and 2/3 minuters.
rate * time = distance becomes rate * (16 and 2/3) = 25
solve for rate to get rate = 25 / (16 and 2/3).
rate is equal to 1.5 miles per minute.
1.5 miles per minute * 60 = 90 miles per hour.
the slope intercxept form of the linear equation is y = mx + b.
m is the slope.
b is the y-intercept.
the point slope form of the linear equation is (y-y1) = m * (x-x1)
m is the slope.
x1,y1 are points on the line.
you can use this formula to find the y-intercept as well, once you know the slope.
the standard form of the linear equation is ax + by = c
a is the coefficient of the x term.
b is the coefficient of the y term.
c is the constant term.
when graphing the equation, you will want to convert the equation to slope intercept form.
that's because that form tells you the slope and y-intercept in one shot.
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