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System of equations word problem: walk & ride

Systems of equations can be used to solve many real-world problems. In this video, we solve a problem about distances walking and riding bus to school.

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  • marcimus pink style avatar for user H. H.
    I get confused by this type of solution because of the units. It's funny how you can just have a system of equations despite all their different units. Like he sums one equation that deals with distance (km) and another that deals with time (hours). How is that not conflicting? This cause my brain to shut down a little...
    (79 votes)
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    • blobby green style avatar for user InnocentRealist
      Maybe it's easier to see this when you make the times, rather than the distances, the unknowns.

      If w is time walking and b is time busing, then

      1) 5w + 60b = 35
      [(5mph * w hrs) + (60mph * b hrs) = 35 miles (that is, speed1*time1 + speed2*time2 = total distance)]

      2) 2w + 2b = 3
      [from w + b = 1.5. (that is, time1 + time2 = total time)].

      You can see here that while equation "1)" is about distance, and equation "2)" about time, that since the rates (5mph and 60mph) are not unknowns, they can be thought of simply as numbers that show another way that the two times are related, which gives you 2 (different) equations in two unknown times, which is enough information to get a solution.
      (43 votes)
  • mr pants teal style avatar for user AnhN
    How did you get 11/12B by adding B and -B/12 together?
    I don't get it at all! Please explain in an easier way..
    (41 votes)
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    • duskpin ultimate style avatar for user Julicz
      according to professionals from NASA and millions of researchers all across the globe:
      We got B and add -B/12 or
      B-B/12
      which is equal to
      1B-1B/12
      look that the Bees have a legitimate exponent corresponding to one(1)
      now:
      1 is and can be 12/12 or any number over its number like 1/1
      so:
      12/12B-1B/12 and 1B/12 is 1/12B
      so:
      12/12B-1/12B
      put bees aside and work out the fraction because everything we're working is with bees:
      (12/12-1/12)B = 11/12 B
      (20 votes)
  • blobby green style avatar for user William Mitchell
    I like to use substitution when solving problems with multiple equations. Is there a preferred method?
    w = time walking, b = time riding the bus
    w + b = 1.5, w = 1.5 - b, b = 1.5 - w
    5w + 60b = 35 (speed times time equals distance)
    5w + 60(1.5 - w)=35
    5w + 90 - 60w = 35
    5w - 60w + 90 - 90 = 35 - 90
    -55w = -55
    w = 1, b = 0.5
    5(1) + 60(.5) = 35
    5 + 30 = 35
    The trip to school: 5 kilometers walked in an hour and 30 kilometers traveled on the bus in 0.5 hours.
    (50 votes)
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  • old spice man green style avatar for user bhannon039
    I struggle when doing word problems in how to define my variables. I noticed in this video that Sal went straight to what question the problem was asking and defined his variables based on that. Is that a good idea in general to define your variables based on the question?
    (25 votes)
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  • leafers ultimate style avatar for user Terrell Tan
    i don't get how you can subtract 1.5hours from 35 kilometers :(
    (19 votes)
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  • mr pants orange style avatar for user Jim Wilson
    SAL moves pretty fast through this video and he does lots of stuff that we've learned before but he doesn't explain in minute detail.

    Let's try to explain what SAL did here.

    The total distance is the sum of the unknown walking distance W and the unknown bus distance B.

    So W + B = 35 km. I will call this the "top equation".

    We know that distance divided by speed is time, so we know that the unknown walking distance W divided by speed of 5km/hr = W distance / 5 kmh and the unknown bus distance is B divided by 60 kmh = B/60.

    So W+B = 35 and W/5 + B/60 = 1.5. I will call this the "bottom equation".

    SAL decides to eliminate the variable "W" and notices that w/5 is 1/5th of W. Remember multiplying fractions? Convert 5 to 5/1, then multiply the top numbers together, then separately multiply the bottom numbers. So W multiplied by 5 is 5W and 5 multiplied by 1 is 5. So W/5 x 5/1 becomes 5W/5. Whenever the top number (numerator) in a fraction and the bottom number are the same number, then the fraction is "1".

    2/2 is 1. 3/3 is 1. 4/4 is one. 5/5 is 1.

    So 5/5 with a W next to it is 1 x W, which is another way of writing "W".

    Sal is trying to eliminate W so what he actually did was multiply by -5, which meant that W/5 x -5/1 is -W.

    To keep the equality, SAL then multiplied the rest of the bottom equation by -5.

    SAL now tackles multiplying B/60 by -5. Remember -5 as a fraction is -5/1. SAL simplified these fractions. Put the two fractions side by side and you simplify the top left by dividing by the bottom right and the bottom left by dividing by the top right. So... look...

    B x -5
    60 1

    60 / -5 = 12. B / 1 is one. That results in -B/12.

    SAL (metaphorically) jumps over the equals sign and multiplies 1.5 by -5 to get -7.5.

    So the system of equations now reads

    W+B=35
    -w-b/12 = -7.5

    SAL now adds the bottom equation to the top equation.

    W-W = 0, so the W disappears!

    B/12 really means "one twelfth of B". 12/12 is a full one e.g. 12/12 of a pie is a full pie, so, just as 12/12 of a pie minus 1/12 of a pie results in 11/12 of a pie, then one full B (or, if you prefer, 12/12 of a B) minus 1/12 of a B = 11/12 of a B i.e. 11/12 x B.

    SAL again jumps over the equality. He adds the bottom to the top... 35 plus -7.5 is 27.5.

    Because we have a fraction in the calculation, SAL converts the decimal number 27.5 into a fraction.

    SAL's pretty clever and spots that 27.5 = 27.5 / 1, and that if he doubles that fraction by 2 he will have the fraction 55 / 2. He does that because he knows that 12 can be divided by 2.

    So the bottom equation now reads 11/12B = 55 / 2.

    To shove the 11/12 fraction from the left to the right, he multiplies it by its reciprocal which is 12/11. 11/12 x 12/11 is one (the 12s cancel each other and the 11s cancel each other).

    That tells him that B = 55/2 x 12/11.

    SAL then cancels these. So 2 / 2 is 1. 12 / 2 is 6. 11 / 11 is one and 55 / 11 is 5. That gives him this...

    5/1 x 6/1. 5 x 6 = 30. 1 x 1 is 1.

    so that means B = 30! That's 30k by bus!

    SAL pus 30 into the "B" spot in the equation W+B=35. So it now reads W+30=35.

    We know that 35-30 is 5. So we now know that W is 5.

    so we know that Yochan walks 5km and takes the bus 30km.

    Personally, I think Yochanan should have just taken a taxi.
    (14 votes)
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  • female robot ada style avatar for user Samantha
    Why does he put w over 5 and b over 60 into fractions? Is there other ways you where you don't have to have fractions to solve the linear equation?
    (6 votes)
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  • blobby green style avatar for user Ruppert Rapport
    Where did the 12 come from? Sounds like it should’ve been
    5b/60.

    5/60 is NOT 12.
    (4 votes)
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  • cacteye green style avatar for user Ally C
    This might be dumb, but why did he multiply the equation by -5?
    (6 votes)
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    • mr pink green style avatar for user David Severin
      No serious question is dumb. In order to use elimination on a system of equations, you have to have one of the variables be additive inverses in the two equations. Since you have a w (or 1w) on the first equation, to eliminate the w's, you need a -1w on the other since w - w = 0. Thus, starting with w/5, multiplying by -5 on whole second equation reaches that goal, adds the negative and 5 * 1/ = 1.
      (8 votes)
  • blobby green style avatar for user Gustavo Flores, Jr.
    Sakura speaks 150 words per minute on average in Hungarian, and 190 words per minute on average in Polish. She once gave cooking instructions in Hungarian, followed by cleaning instructions in Polish. Sakura spent 5 minutes total giving both instructions, and spoke 270 more words in Polish than in Hungarian.

    This was very confusing how the problem gave the step by steps. I don't understand what made the y value a negative along with the constant after the equal symbol. the step by step wrote it as 150x-190y=-270, I thought it would be 150x+190y=270 And x+y=5
    (6 votes)
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    • mr pink green style avatar for user David Severin
      Let P be time in Polish and H be time in Hungarian, so P + H = 5
      I would have started the second equation as 190P = 150H + 270 (more than is + ). Thus it could be either 190P - 150H = 270 (this is the one that makes more sense to me) or 150H - 190P = -270 (this would work, but not what I would choose).
      I would then multiply first times 150, 150P + 150P = 750, add two equations 340P = 1020, P = 3 so H = 2. Thus, 190(3) = 570 and 150(2) = 300 to check our second equation 570 = 300 + 270.
      Hope this helps your understanding.
      (5 votes)

Video transcript

- [Voiceover] Yochanan walked from home to the bus stop at an average speed of five kilometers per hour. He immediately got on his school bus and traveled at an average speed of 60 kilometers per hour until he got to school. The total distance from his home to school is 35 kilometers, and the entire trip took one and a half hours. How many kilometers did Yochanan cover by walking, and how many kilometers did he cover by travelling on the bus? It's fascinating. So I encourage you to pause the video and try to think about on your own. Alright, 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 travelling 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 can actually 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 and see they tell us, they tell us that it is 35 kilometers to school so this is his school right over here. I'll draw a bigger building, that's his school, and we know that this distance is 35, 35 kilometers and we also know that it took one and a half hours. One and a half hours, now he traveled at different rates for different distances. So he 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 the rest of the distance, he covered by the bus, so the rest of this distance, all of this distance 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, 35 kilometers here, this is the entire, that is the entire distance 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 half hours, so we'll just write that over here. This is going to be 1.5, so what's the time traveled by, what's the time he walks? Let me write this over here, time time walking, we'll that's going to be the distance walking divided by the rate walking. So the distance walking is W kilometers W kilometers divided by his rate, the distance divided by your rate is gonna give you your time, so let's see, his rate is five kilometers per hour, five kilometers per hour and so you're gonna have kilometers cancel kilometers and if you divide by or if you have one over hours in the denominator, that's going to be the same thing, this is gonna be W over five hours, so the units work out. So his time walking is W over five, W over five 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 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 five, and I'm gonna, this is going to be a negative W here so it'll cancel out with this W up there. So let's do that, let's multiply the second equation by, I'm just gonna 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, I'm going to multiply both sides by negative five I'm going to get negative five times W over five is negative W, negative five times B over 60. Let's see, it's gonna be, it's going to be negative five over 60, that's negative 1/12, so this is negative B over 12 and then it is going to be equal to 1.5 times negative five is negative 7.5, negative 7.5. Now we can just add the left and right hand sides of these two equations. Now let me, I can do this a little bit neater, let me actually delete, let me make these line up a little bit better so that we, delete that, make this, so this first equation was, whoops, this first equation was W plus B is equal to 35 now they line up better and now we can add the left hand 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 the twelfth of B, well that's going to be 11/12B, 11/12B is equal to, let's see, 35 minus 7.5, see 35 minus seven would be 28 and then another half this would be 27.5. 27.5 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. Let's 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 two in the denominator so I can make that a six and a one, then I have a 55 and 11, I can divide both by 11, so it's gonna be five and a one. So it's five times six, B is equal to 30 and B was in kilometers so if he travels 30 kilometers, 30 kilometers by bus, B is equal to 30 kilometers, and the amount that he walks, well, we can figure that out. If this is 30, well the amount that he walks is going to be five. This is going to be five kilometers, W, let me write this, W is equal to five kilometers. He walks five kilometers and then he goes by the bus 30 kilometers.