Systems of equations word problems
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Example 4: Solving a word problem with substitution
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Mixture problems 1
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Mixture problems 2
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Mixture problems 3
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Systems and rate problems
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Systems and rate problems 2
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Systems and rate problems 3
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Officer on Horseback
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Two Passing Bicycles Word Problem
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Passed Bike Word Problem
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Passing Trains
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Overtaking Word Problem
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Problem Solving Word Problems 2
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Systems of equations word problems
Systems and rate problems 2 Systems and rate problems 2
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- Joelle has two after-school jobs, babysitting and working
- at her parents' grocery store.
- Working at the grocery pays her $12 an hour, and
- babysitting pays her $9.50 an hour.
- Last week she worked 19 hours in total.
- That looks like a statement that
- could lead to an equation.
- Last week she worked 19 hours in total.
- And she earned $56 more at the grocery store than she did
- babysitting.
- That looks like another statement
- that could be an equation.
- She earned $56 more at the grocery store than she did
- babysitting.
- And then they ask us, how many hours did she
- work at each job?
- So they want to know-- so let's define
- some variables here.
- Let's define time babysitting, so t sub b, as equal to hours
- babysitting.
- And let's define t sub g as hours at the grocery.
- So they want us to figure out what was t sub b and t sub g,
- what were the hours of babysitting and what were the
- hours at the grocery?
- Now, they tell us that she worked 19 hours total.
- So that blue statement says that the sum of these two is
- going to be 19.
- So the time she worked babysitting plus the time she
- worked at the grocery store is going to be equal to 19 hours.
- Now the other statement.
- This by itself obviously we can't solve for these.
- We need another constraint.
- They said that she earned $56 more at the grocery store then
- she did babysitting.
- So let me define two more variables here.
- So this is the pay babysitting, her total pay
- babysitting.
- And let's say P sub g is equal to the pay
- at the grocery store.
- Now, what are these terms?
- I mean, if we just look at this second statement right
- there, we'll just say, OK, the pay at babysitting, she earned
- $56 more at the grocery than she did babysitting.
- So this second statement tells us that, well, I guess the
- pay-- let me write it over here-- the pay babysitting,
- she earned $56 more at the grocery than she did
- babysitting.
- So if we take babysitting and add $56, that will get us to
- how much she earned at the grocery store.
- That's what that second statement is telling us. $56
- more at the grocery than babysitting.
- So if you add these two, you get the grocery.
- Now, this still isn't cool.
- We have one, two, three, four unknowns
- and only two equations.
- That's not going to help us.
- But what will help us we can express these two variables in
- terms of these two variables, and then we have two equations
- with two unknowns, so we can solve them the way we've
- solved all systems of equations.
- Now the other information they give us is the rate, how much
- she earned per hour.
- Working at the grocery, she makes $12 per hour.
- Babysitting, she makes $9.50 per hour.
- So what is going to be her pay babysitting?
- Let me write this over here.
- Her pay babysitting-- I'll do it in this color-- her pay
- babysitting is going to be equal to the time
- babysitting, right?
- The time babysitting times how much she gets paid per hour.
- The time babysitting is how many hours, times how much she
- gets paid per hour.
- It tells us right there.
- She gets paid $9.50 per hour babysitting.
- So times $9.50.
- That is what her pay babysitting is.
- And then, same argument, what's her pay working at the
- grocery store?
- Her total pay working at the grocery store.
- Her pay at the grocery is going to be equal to the time
- at the grocery times the rate that she gets paid at the
- grocery. $12 per hour.
- So times $12 per hour.
- And this hopefully is intuitive.
- If I'm making $12 per hour and I work for, I don't know, 3
- hours, I'm going to make $36.
- I would just multiply these two.
- If I worked 2 hours at $9.50, I'm going to make $19.
- That's all this is.
- A total pay.
- So now we can take these two expressions-- we could take
- this expression and this expression-- and substitute it
- back into that equation right there, and then we'll have an
- equation that's only in terms of the time babysitting and
- the time at the grocery store.
- So let's do that.
- So this first term right here, her total pay babysitting,
- that is this expression right there.
- So let me write it right here.
- So $9.50 times the time babysitting.
- So I'll just write 9.5 times time babysitting-- that's this
- term right here-- plus 56 is equal to the pay that she got
- at the grocery, which is 12 times the time at the grocery.
- And now we need to solve for both of these variables.
- We have two equations and two unknowns.
- This is just straightforward algebra at this point.
- So let's do it.
- So the easiest way to do this might just be to do straight
- up substitution cause we have all these crazy numbers here.
- Let's just solve for the time at the babysitting in terms of
- the time at the grocery.
- So if you take this top equation-- let me write it
- over here-- so the time babysitting plus the time at
- the grocery is equal to 19.
- If you subtract the time at the grocery from both sides,
- you get the time babysitting is equal to-- these cancel
- out-- 19 minus the time at the grocery.
- So we solved for the time babysitting in terms of the
- time at the grocery.
- So everywhere we see the time babysitting, we can substitute
- it with this expression right here.
- That's what t sub b is.
- That is what the time babysitting is.
- So let's do that.
- So this equation becomes 9.5 times the time babysitting,
- which we just figured out from this top equation is 19 minus
- the time at the grocery, plus 56, is equal to 12 times the
- time at the grocery.
- And now we just have to solve for the time at the grocery.
- So 9.5 times 19.
- That's a little bit of hairy.
- Let me just do it over here, on the right side right here.
- So if I have 19 times 9.5.
- 5 times 9 is 45.
- 1 times 5 is 5, plus 4, is 9.
- And then we have a 0 here.
- 9 times 9 is 81.
- 9 times 1 is 9, plus 8, is 17.
- So 5 plus 0 is 5.
- 9 plus 1 is 10.
- 1 plus 7 is 8.
- You just have a 1 there, and you have one number behind the
- decimal point.
- So it's 180.5.
- So 9.5 times 19 is 180.5 minus 9.5 times the time at the
- grocery, plus 56, is equal to 12 times
- the time at the grocery.
- Now what's 180.5 plus 56?
- So 180 plus 50-- let's add just those two terms-- 180.5
- plus 56, just to simplify it.
- 180 plus 50 would be 230.
- So this'll be 236.5 minus 9.5 times the time at the grocery,
- is equal to 12 times the time at the grocery.
- Now we can add the time at the grocery to both sides of this
- equation, or the 9.5.
- So let's do that.
- Plus 9.5 times the time at the grocery is equal-- or plus 9.5
- times time at the grocery.
- Just adding it to both sides of that equation.
- And we are left with-- the left-hand side, that cancels
- out-- we just have 236.5 is equal to, 12 plus 9 is 21, so
- this is going to be 21.5.
- It's equal to 21.5 times the time she spent at the grocery.
- So now we can just divide both sides of this equation by
- 21.5, and we've figured out how long she
- spent at the grocery.
- Let's do that.
- Divide both sides by 21.5.
- So that cancels out.
- The time she spent at the grocery is going to be equal
- to that thing.
- And let's see what that is.
- We get 21.5 goes into 236.5.
- Let's multiply both of these numbers by 10.
- So this is the same thing as 215 going into 2,365.
- The decimal goes right there.
- 215 goes into 236 one time.
- 1 times 215 is 215.
- And then we can subtract-- let me scroll down more-- this
- problem is hairier than I thought it would be.
- So 36 minus 15 is 21.
- So 21-- bring down the 5-- and lucky for us that's pretty
- straightforward.
- 215 goes into 215 one time.
- 1 times 215 is 215.
- Subtract.
- You get no remainder.
- So the time at the grocery store is 11, or 11 hours.
- Now we can go and substitute back to figure out her time
- babysitting.
- We know that the time babysitting is equal to 19
- minus the time at the grocery.
- In this case, it's 19 minus 11.
- We just figured that out.
- 19 minus 11 is 8.
- So her time babysitting, she spent 8 hours babysitting.
- And we are done.
- They both add up to 19, and you could make sure that they
- also verify.
- That also works out for the second equation.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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