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Ex: Work Linear Function Application (Slope, Intercept Meaning) - YouTube
Channel: Mathispower4u
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you have 845 decorative bricks delivered
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to your house and set down in your
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driveway
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you need to move all the bricks to your
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backyard you move a few bricks by hand
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and quickly decide that that will take
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forever so you go buy a wheelbarrow to
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haul the rest you decide to pace
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yourself by hauling five wheelbarrows
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full of bricks each hour
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after four hours you determine you have
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moved 245 bricks to the backyard after
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seven hours you have moved 425 bricks to
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the backyard the first question is how
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many bricks an hour are you moving using
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the wheelbarrow and this value is
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actually the slope of our linear
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equation
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where if we have a linear equation in
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slope-intercept form
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it fits the form y equals mx plus b
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where m
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is the slope which is the constant rate
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of change or in this case the number of
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bricks moved per hour
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b is the vertical intercept where the
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value when the initial input or in this
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case x is equal to zero
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so going back to the given information
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first thing to notice is because we're
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concerned about the number of bricks per
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hour we really don't care that you're
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hauling five wheelbarrows full of bricks
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per hour
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we only care that after four hours you
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determine
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245 bricks have been moved and after
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seven hours
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you determine 425 bricks have been moved
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let's first put this information in
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ordered pairs
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so the first point would be four comma
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245
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because after four hours
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245 bricks have been moved or we can say
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when the input is four hours the output
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is 245 bricks moved
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and after seven hours
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425 bricks have been moved so the second
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ordered pair would be seven comma four
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hundred twenty-five
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now for this problem we're going to be
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finding a linear function in the form of
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q of t
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so we can say the input would be t the
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first coordinate
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and the output would be q of t
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the quantity of bricks moved
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now that we have the two ordered pairs
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we can find the slope or the number of
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bricks moved per hour using the
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wheelbarrow
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by determining the change in the
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function value divided by the change in
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the input or in our case we would have
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the slope equals the change in q or q of
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t
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divided by the change of t
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so for the change of q we'd have 425
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minus
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245 and for the change of t we'd have
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seven
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minus four
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so we have 180
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divided by three
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which equals 60 which means you're
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moving
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60 bricks
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per hour
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using the wheelbarrow
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so your answer would be i am moving 60
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bricks an hour using the wheelbarrow
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again let's go ahead and record over
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here that we have m
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the slope equals 60 or if we want 60
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bricks per hour
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now the second question asked how many
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bricks had you moved by hand before you
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started using the wheel barrel this
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value is actually going to be the
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vertical intercept often known as the
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y-intercept of our linear equation we'll
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find this algebraically in the next part
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let's try to find it logically
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if we now know that you're moving 60
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bricks per hour and after 4 hours you've
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moved 245 bricks
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we can use this information to work
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backwards to determine how many bricks
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you moved at the start which would be at
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zero hours
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so we'll go ahead and use this first
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point
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we could use the second one but let's go
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ahead and use the first one
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and the fact that we know the slope is
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equal to 60.
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so if we wanted to determine how many
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bricks you had moved at 3 hours
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we'd have to subtract 60 one time
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if we wanted to determine how many
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bricks you had moved after two hours
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we'd have to subtract 60 two times and
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so on so if our goal is to determine
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the number of bricks moved when time
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equals zero
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or before you started using the
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wheelbarrow
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we could take the amount moved at 4
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hours 245
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and subtract 4 x 60
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to determine
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how many bricks moved at time 0. again
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we're subtracting four times 60 because
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245 bricks are removed after four hours
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we want to know how many removed after
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zero hours so we have to subtract
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four 60s
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so this would be 245
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minus 240
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which equals 5. so your answer would be
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i moved five bricks by hand
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before i started using the weber
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now the next part asks us to write the
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linear equation that represents this
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situation we're asked to write the
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equation in the form of q of t equals mt
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plus b
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where we already know q of t is the
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quantity of bricks moved after t hours
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using the wool barrel
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well we already know that m the slope is
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equal to 60 and we actually already know
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the vertical intercept is five
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but let's just assume we didn't know the
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vertical intercept was five we could set
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this up as q of t
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equals
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60t plus b
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and now we could use one of our points
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to determine the value of b
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again remember the first coordinate is t
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and the second coordinate is q of t
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we could use either point but to see the
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connection from the previous part let's
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use this first point again
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so we'll substitute 245 for q of t
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and four for t
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so we'd have 245
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equals 60 times four plus b
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this looks very similar to the last part
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we have 245
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equals 240
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plus b
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subtracting 240 on both sides
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we can see the vertical intercept is
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five
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so if we know the slope equals 60 and
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the vertical intercept equals five
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we'd have q of t equals 60t
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plus five
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and now for the last part we want to
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find the value of t
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or how many hours it's going to take
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until q of t the quantity of bricks
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equals 845. remember this is the total
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number of bricks that you had delivered
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so what we'll do
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is using a linear equation here we'll
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substitute 845 for q of t
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and solve for t
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so we'd have 845
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equals 60 t plus five
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subtract five on both sides that would
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give us 840
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equals 60t
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divide both sides by 60.
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so we have t equals
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14.
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remember this would be in hours we're
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asked to write our answer as an ordered
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pair so the ordered pair would be open
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parenthesis t which is 14
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comma q of 14 which is 845.
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remember as an ordered pair
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the input is always first and the output
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is always second or in this case t is
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first and q of t is second and now we're
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asked to complete the following sentence
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after using the wheelbarrow four 14
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hours
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i will have moved all the bricks to the
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backyard
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i hope you found this helpful
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