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CCSS.Math: , , , ,

yochanan walked from home to the bus stop at an average speed of 5 km/h he immediately got on his school bus and traveled at an average speed of 60 km/h until he got to school the total distance from his home to school is 35 kilometers in the entire trip took one and a half hours how many kilometers did Jochen uncover by walking and how many kilometers did he cover by traveling on the bus is fascinating so I encourage you to pause the video and try to think about it on your own all right so let's just define some variables here how many kilometers did he cover by walking let's call this w and how many kilometers did he cover by traveling on the bus let's call that let's call that B and so what do we know we know that the kilometers and I actually can even draw it just to make sure that we're visualizing this thing right so this right over here is his home that's his home and then he is going to travel let's see they tell us they tell us that is 35 kilometers to school so this is a school right over here I'll draw a bigger building that's his school and we know that this distance is 35 35 kilometers we also know that it took one and a half hours one and a half hours I know he traveled different rates for different distances so you traveled some distance to the school bus so this is or to the bus stop so that's the bus stop right over there and we're seeing this distance to the bus stop that's how much he covered by walking so this right over here this distance right over here that is W and then the rest of the distance he covered by the bus so the rest of this distance all of this business right over there that is going to be B so what do we know we know the distance covered by walking plus the distance covered by bus is going to be 35 kilometers remember the 35 kilometers here this is the entire that is the entire distance from home from home to school so we know that W plus B W plus B plus B is equal to 35 kilometers is equal to 35 kilometers and just with one equation we're not going to be able to figure out what W and B are but we have another constraint we know the total amount of time so the total amount of time is going to be one and a half hours so let me just write that over here this is going to be one point five so what's the time traveled by - what's the time he walks let me write this over here time time walking well that's going to be the distance walking divided by the rate walking so the distance walking is W kilometres W kilometres divided by his rate distance divided by your rate is going to give you your time so let's see his rate is 5 km/h 5 km/h and so you're gonna have kilometers cancel with kilometers and if you divide by or if you have 1 over hours in the denominator that's going to be the same thing this is going to be W over 5 hours so the units work out so his time walking is W over 5 W over 5 and by that same logic his time on the bus is going to be the distance on the bus divided by divided by the average speed of the school bus so this is going to be 60 this is all going to be in hours and now we can solve this system of equations we have two two linear equations with two unknowns we should be able to find W and B that satisfy both of these now what's an easy thing to do let's see if I can multiply this second equation by negative negative five that I'm going to this is going to be a negative W here and so it'll cancel out with this W up there so let's do that so let's multiply the second equation by and I'm just going to switch to one color here so this top equation is going to be W plus B is equal to 35 this bottom equation if I multiply both sides by negative five so both sides by negative five so I'm going to multiply both sides by negative five I'm going to get negative five times W over 5 is negative W negative 5 + x be over 60 let's see it's going to be negative five over 60 that's negative 112 so because it's negative be over 12 and then is going to be equal to 1.5 times negative 5 is negative seven point five negative seven point five now we can just add the left and right hand sides of these two equations and let me I could do this a little bit neater let me actually delete let me make these line up a little bit better so let me delete that make this so this this first equation was whoops this first equation was w plus B is equal to 35 now they line up better and now I can have the left side sides of these equations and the right-hand sides so the left-hand sides the W's cancel out that's what we wanted now B minus B over 12 or B minus 1/12 of B well that's going to be 1112 to be 1112 B is equal to let's see 35 minus 7.5 see 35 minus 7 would be 28 and then another half this would be twenty seven point five twenty seven point five and since I'm dealing with a fraction here let me write this as a fraction so this is the same thing as 55 over two so let me just write it this way this is the same thing as 55 over two now to solve for B I just have to multiply both sides times this reciprocal I'll switch colors just to ease the monotony so multiply both sides by 12 over 11 12 over 11 what we get these cancel out what we get is that B so I'll do this in this color B is equal to let's see I have a 12 in the numerator 2 in the denominator so I can make that a 6 and a 1 and then I have a 55 and 11 I could divide both by 11 so it's going to be 5 and a 1 so it's 5 times 6 B is equal to 30 and B was in kilometers so he travels 30 kilometers 30 kilometers by bus B is equal to 30 kilometers and the amount that he wha well we could figure that out if this is 30 well the amount that he walks is going to be 5 this is going to be 5 kilometers w let me write this w is equal to 5 kilometers he walks 5 kilometers and he goes by the bus 30 kilometers