# The price of a stereo at Mary's is \$220 less than twice the price at Cal's. The difference in the price between both stores is \$175. How much does the system cost in both stores?

Let us assign variables: let m be the price of the system at Mary's and let c be the price of the system at Cal's.

We are told that the price at Mary's, m, is \$220 less than twice the price at Cal's. Translating into algebraic notation we get m=2c-220. (Twice indicates to multiply by 2, while in context less suggests we subtract.)

Also we are given that the difference (subtraction) between the two prices is \$175 or m-c=175.

We have a system of two linear equations in two unknowns. This system might have no solutions (inconsistent), an infinite number of solutions (consistent and dependent), or exactly one solution (consistent and independent.)

We can solve the system in a number of ways: We could use a guess and revise system (very inefficient), we could graph the system in a coordinate plane (inefficient and difficult to get an exact answer), or, algebraically, we could use substitution or linear combinations. If we know about matrices we could use Cramer's method, Gaussian elimination, or inverse matrices.

Here we can easily use substitution:

m=2c-220
m-c=175 `rArr` m=175+c

We have two expressions for m so we can set these equal to one another to get:

2c-220=175 + c
c=395 (subtract c from both sides and add 220 to both sides.)

Then m=175+395=570.

The solution: the price at Mary's is \$570 while at Cal's it is \$395.

Checking we see that 570-395=175 so the system prices are 175 dollars apart. Also, 2(395)-220=790-220=570 so the price at Mary's is \$220 less than twice the price at Cal's.