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Joelle has two after-school jobs, babysitting and working

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at her parents' grocery store.

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Working at the grocery pays her $12 an hour, and

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babysitting pays her $9.50 an hour.

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Last week she worked 19 hours in total.

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That looks like a statement that

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could lead to an equation.

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Last week she worked 19 hours in total.

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And she earned $56 more at the grocery store than she did

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babysitting.

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That looks like another statement

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that could be an equation.

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She earned $56 more at the grocery store than she did

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babysitting.

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And then they ask us, how many hours did she

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work at each job?

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So they want to know-- so let's define

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some variables here.

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Let's define time babysitting, so t sub b, as equal to hours

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babysitting.

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And let's define t sub g as hours at the grocery.

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So they want us to figure out what was t sub b and t sub g,

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what were the hours of babysitting and what were the

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hours at the grocery?

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Now, they tell us that she worked 19 hours total.

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So that blue statement says that the sum of these two is

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going to be 19.

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So the time she worked babysitting plus the time she

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worked at the grocery store is going to be equal to 19 hours.

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Now the other statement.

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This by itself obviously we can't solve for these.

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We need another constraint.

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They said that she earned $56 more at the grocery store then

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she did babysitting.

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So let me define two more variables here.

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So this is the pay babysitting, her total pay

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babysitting.

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And let's say P sub g is equal to the pay

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at the grocery store.

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Now, what are these terms?

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I mean, if we just look at this second statement right

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there, we'll just say, OK, the pay at babysitting, she earned

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$56 more at the grocery than she did babysitting.

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So this second statement tells us that, well, I guess the

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pay-- let me write it over here-- the pay babysitting,

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she earned $56 more at the grocery than she did

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babysitting.

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So if we take babysitting and add $56, that will get us to

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how much she earned at the grocery store.

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That's what that second statement is telling us. $56

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more at the grocery than babysitting.

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So if you add these two, you get the grocery.

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Now, this still isn't cool.

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We have one, two, three, four unknowns

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and only two equations.

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That's not going to help us.

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But what will help us we can express these two variables in

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terms of these two variables, and then we have two equations

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with two unknowns, so we can solve them the way we've

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solved all systems of equations.

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Now the other information they give us is the rate, how much

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she earned per hour.

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Working at the grocery, she makes $12 per hour.

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Babysitting, she makes $9.50 per hour.

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So what is going to be her pay babysitting?

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Let me write this over here.

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Her pay babysitting-- I'll do it in this color-- her pay

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babysitting is going to be equal to the time

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babysitting, right?

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The time babysitting times how much she gets paid per hour.

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The time babysitting is how many hours, times how much she

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gets paid per hour.

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It tells us right there.

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She gets paid $9.50 per hour babysitting.

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So times $9.50.

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That is what her pay babysitting is.

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And then, same argument, what's her pay working at the

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grocery store?

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Her total pay working at the grocery store.

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Her pay at the grocery is going to be equal to the time

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at the grocery times the rate that she gets paid at the

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grocery. $12 per hour.

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So times $12 per hour.

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And this hopefully is intuitive.

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If I'm making $12 per hour and I work for, I don't know, 3

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hours, I'm going to make $36.

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I would just multiply these two.

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If I worked 2 hours at $9.50, I'm going to make $19.

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That's all this is.

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A total pay.

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So now we can take these two expressions-- we could take

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this expression and this expression-- and substitute it

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back into that equation right there, and then we'll have an

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equation that's only in terms of the time babysitting and

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the time at the grocery store.

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So let's do that.

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So this first term right here, her total pay babysitting,

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that is this expression right there.

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So let me write it right here.

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So $9.50 times the time babysitting.

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So I'll just write 9.5 times time babysitting-- that's this

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term right here-- plus 56 is equal to the pay that she got

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at the grocery, which is 12 times the time at the grocery.

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And now we need to solve for both of these variables.

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We have two equations and two unknowns.

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This is just straightforward algebra at this point.

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So let's do it.

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So the easiest way to do this might just be to do straight

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up substitution cause we have all these crazy numbers here.

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Let's just solve for the time at the babysitting in terms of

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the time at the grocery.

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So if you take this top equation-- let me write it

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over here-- so the time babysitting plus the time at

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the grocery is equal to 19.

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If you subtract the time at the grocery from both sides,

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you get the time babysitting is equal to-- these cancel

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out-- 19 minus the time at the grocery.

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So we solved for the time babysitting in terms of the

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time at the grocery.

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So everywhere we see the time babysitting, we can substitute

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it with this expression right here.

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That's what t sub b is.

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That is what the time babysitting is.

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So let's do that.

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So this equation becomes 9.5 times the time babysitting,

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which we just figured out from this top equation is 19 minus

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the time at the grocery, plus 56, is equal to 12 times the

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time at the grocery.

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And now we just have to solve for the time at the grocery.

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So 9.5 times 19.

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That's a little bit of hairy.

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Let me just do it over here, on the right side right here.

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So if I have 19 times 9.5.

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5 times 9 is 45.

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1 times 5 is 5, plus 4, is 9.

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And then we have a 0 here.

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9 times 9 is 81.

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9 times 1 is 9, plus 8, is 17.

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So 5 plus 0 is 5.

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9 plus 1 is 10.

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1 plus 7 is 8.

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You just have a 1 there, and you have one number behind the

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decimal point.

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So it's 180.5.

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So 9.5 times 19 is 180.5 minus 9.5 times the time at the

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grocery, plus 56, is equal to 12 times

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the time at the grocery.

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Now what's 180.5 plus 56?

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So 180 plus 50-- let's add just those two terms-- 180.5

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plus 56, just to simplify it.

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180 plus 50 would be 230.

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So this'll be 236.5 minus 9.5 times the time at the grocery,

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is equal to 12 times the time at the grocery.

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Now we can add the time at the grocery to both sides of this

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equation, or the 9.5.

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So let's do that.

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Plus 9.5 times the time at the grocery is equal-- or plus 9.5

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times time at the grocery.

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Just adding it to both sides of that equation.

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And we are left with-- the left-hand side, that cancels

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out-- we just have 236.5 is equal to, 12 plus 9 is 21, so

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this is going to be 21.5.

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It's equal to 21.5 times the time she spent at the grocery.

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So now we can just divide both sides of this equation by

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21.5, and we've figured out how long she

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spent at the grocery.

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Let's do that.

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Divide both sides by 21.5.

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So that cancels out.

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The time she spent at the grocery is going to be equal

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to that thing.

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And let's see what that is.

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We get 21.5 goes into 236.5.

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Let's multiply both of these numbers by 10.

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So this is the same thing as 215 going into 2,365.

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The decimal goes right there.

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215 goes into 236 one time.

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1 times 215 is 215.

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And then we can subtract-- let me scroll down more-- this

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problem is hairier than I thought it would be.

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So 36 minus 15 is 21.

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So 21-- bring down the 5-- and lucky for us that's pretty

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straightforward.

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215 goes into 215 one time.

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1 times 215 is 215.

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Subtract.

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You get no remainder.

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So the time at the grocery store is 11, or 11 hours.

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Now we can go and substitute back to figure out her time

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babysitting.

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We know that the time babysitting is equal to 19

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minus the time at the grocery.

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In this case, it's 19 minus 11.

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We just figured that out.

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19 minus 11 is 8.

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So her time babysitting, she spent 8 hours babysitting.

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And we are done.

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They both add up to 19, and you could make sure that they

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also verify.

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That also works out for the second equation.

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