System of equations word problem Andrea's Sequoia gets 12 miles per gallon while driving in the city, and 20 miles per gallon while on the highway. The last time she filled up the car it took 16 gallons of gas, and she had driven 280 miles since the previous fill-up. How many miles of that 280 were city driving?
 A: This is a standard type of problem where you're given enough information to form two linear equations, with two unknowns. And the process outlined below is one that will generalize to many such situations. So this is a good time to learn and understand how you can approach such a problem:

Let $\,c =$ (the number of miles that Andrea drove in the city since her last fill-up).    
Let $\,h = $ (the number of miles she drove on the highway since her last fill-up.) 


*

*We know Andrea drove a total of $280$ miles since her last fill-up, so $c$ and $h$ must add up to $280$.
So one equation relating $c$ and $h$ is 



$$c + h = 280\tag{1}$$



*

*We are also given Andrea gets $\,12\,$ miles per gallons while citi-driving, which means she used $\dfrac c{12}$ gallons driving in the city since her last fill up. And since she gets $20$ miles per gallon on the highway, she must have used $\dfrac h{20}$ gallons for highway-driving since her last fill-up. And we're given since her last fill-up, she used a total of $16$ gallons. So the gallons used while city-driving and the gallons used while highway-driving must add up to $16$. So our second equation relating $c$ and $h$ is given by:



$$\frac c{12} + \frac h{20}=16 \tag{2}$$

From equation $(1)$, we can express miles of city driving, $c$ as follows: $$c + h = 280 \implies c = 280 - h.\tag{c}$$
Now simply substitute $c = 280 - h$ into $c$ of equation $(2)$, and solve for $h$. Then you can easily determine what $c$ must be by using the equation for $(c)$.
A: HINT: Start by assigning variables to the unknown quantities. Let $x$ be the number of miles that she drove in the city since the previous fill-up, and let $y$ be the number of miles that she drove on the highway in that period. Since she gets $12$ miles per gallon while driving in the city, she must have used $\frac{x}{12}$ gallons while driving in the city. Similarly, she must have used $\frac{y}{20}$ gallons on the highway. She used a total of $16$ gallons, so we have one equation:
$$\frac{x}{12}+\frac{y}{20}=16\;.$$
The other equation is easy to find: she drove a total of $280$ miles in that period, so ... ?
Now you have two linear equations in the unknowns $x$ and $y$, and you can solve them by any of the standard methods.
(It’s possible to set up the problem directly with only a single unknown, say the $x$ of the solution outlined above, but if you’re having difficulty with the setup it’s probably easier to start with one variable for each unknown quantity and then eliminate one.)
