Question 174973
Hi, Hope I can help,
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I need help!!
- A car travelled 470 km from Sudbury to Brandford in 5 hours. For part of the trip, the car travelled at 100 km/h. For the rest of the trip, it travelled at 90 km/h. How far did the car travel at each speed? 
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We will start fresh
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First we need to know a formula
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{{{ (rate)(time)= distance }}}, or with abbreviations, {{{ r)(t) = d }}}
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A car travelled 470 km from Sudbury to Brandford in 5 hours. For part of the trip, the car travelled at 100 km/h. For the rest of the trip, it travelled at 90 km/h. How far did the car travel at each speed? 
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This problem is a little tricky, since the car is going two speeds, instead of just one, we also have two different times.
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This will mean we will have two unknowns, we can find the other unknowns with the information we have
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A car travelled 470 km from Sudbury to Brandford in 5 hours. For part of the trip, the car travelled at 100 km/h. For the rest of the trip, it travelled at 90 km/h. How far did the car travel at each speed?
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Remember that {{{ rate(time) = distance }}}, we don't know the time or distance for either one of the speeds, but we know the total distance and total time
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We can say that the time it took when the car went 90 km/h, the time it took would be {{{ x }}}
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We can say that how long it took when the car went 100 km/h, the time it took would be {{{ 5 - x }}}, ( if you subtract the other time, from the total time, you will find the second unknown time )
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It is the same concept with the distances
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one distance would be {{{ y }}}
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the second unknown distance would be {{{ 470 - y }}}
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We can now find equations to this problem
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{{{ r(t) = d }}}
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Equation 1
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if the car went 100 km/h ( rate )
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if the car took {{{ 5 - x }}} hours ( time ), ( doesn't matter which time we could use "x" if we wanted, but we will use this one )
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These two multiplied together will equal a distance {{{ 470 - y }}} , ( once again it doen't matter which distance, we could use "y" if we wanted )
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Now we would put the words into a formula,
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{{{ rate(time)= distance }}} = {{{ 100(5 - x) = 470 - y }}}, this is the first equation
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Now we will use the other distance, time, and rate, to find the second equation
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If the car went 90 km/h ( rate )
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If the car took {{{ x }}} hours ( time )
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If the car drove {{{ y }}} km ( distance )
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Now replace "r", "t" and "d" with the other veriables
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{{{ r(t) = d }}} = {{{ 90(x) = y }}}, this is the second equation
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Now we can put these two equations side by side
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{{{ system ( 100(5 - x) = 470 - y, 90(x) = y) }}}
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To make it easier, we will simplify the two equations
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Equation 1, {{{ 100(5 - x) = 470 - y }}}, we can use the distribution to simplify this equation
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{{{ 100(5 - x) = 470 - y }}} = {{{ highlight(100)(highlight(5) - x) = 470 - y }}} = {{{ highlight(100)(5 - highlight(x)) = 470 - y }}}
(remember the signs
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{{{ 100(5) - 100(x) = 470 - y }}} = {{{ 500 - 100x = 470 - y }}}, we don't have to move anything right now, so this is the simplified form
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Equation 2, we will just multiply the "90" and "x"
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{{{ 90(x) = y }}} = {{{ 90x = y }}}, this is simplified
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Now we can start solving for the two unknowns
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{{{ system ( 500 - 100x = 470 - y, 90x = y) }}}
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This is a system of equations, the easiest way to solve this system is by substitution
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Since {{{ y = 90x }}}, we can just simply replace "y" in the first equation with "90x"
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{{{ 500 - 100x = 470 - y }}} = {{{ 500 - 100x = 470 - (90x) }}}
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{{{ 500 - 100x = 470 - (90x) }}} = {{{ 500 - 100x = 470 - 90x }}}, now we can just solve for "x", we will move (-100x) to the right side
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{{{ 500 - 100x = 470 - 90x }}} = {{{ 500 - 100x + 100x = 470 - 90x + 100x }}} = {{{ 500 = 470 + 10x }}}, now we will move "470" to the left side
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{{{ 500 = 470 + 10x }}} = {{{ 500 - 470 = 470 - 470 + 10x }}} = {{{ 30 = 10x }}}, to find "x" we will divide each side by "10"
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{{{ 30 = 10x }}} = {{{ 30/10 = 10x/10 }}} = {{{ 3 = highlight(10)x/highlight(10) }}} = {{{ 3 = cross(10)x/cross(10) }}} = {{{ 3 = x }}}, {{{ x = 3 }}}
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(remember "x" is the time in hours), we can replace "x" with "3" in the second equation, to find "y"
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{{{ 90x = y }}} = {{{ 90(3) = y }}} = {{{ 270 = y }}}, {{{ y = 270 }}}, we can check our two answers by replacing "x" and "y" in the first equation
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x = 3
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y = 270
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{{{ 500 - 100x = 470 - y }}} = {{{ 500 - 100(3) = 470 - (270) }}} = {{{ 500 - 300 = 470 - 270 }}} = {{{ 200 = 200 }}} (True)
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Here are the simplified equations with the numbers
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{{{ system ( 100(5 - 3) = 470 - 270, 90(3) = 270) }}}
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{{{ system ( 100(2) = 200, 90(3) = 270) }}}
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If we look at the formula, {{{ rate(time) = distance }}}
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{{{ system ( 100(2) = 200, 90(3) = 270) }}}
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This means that the car went 200 km at 100 km/h, and it went 100 km/h for 2 hours
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The car went 270 km at 90 km/h, and it went 90 km/h for 3 hours
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The distances add up to the total distance, as well as the time with the total time
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The car went 200 km at 100 km/h
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The car went 270 km at 90 km/h
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Hope I helped, Levi