Math 121 Hints & Examples
Word Problems with Two Variables
Word problems that can be solved with two
variables (this implies that there will be two equations)
Example:
The sum of two numbers is 81. The difference of twice
one number and three times the other is 62. Find the two numbers.
First define the variables: X = number 1, Y= number 2
Solution:
The first sentence says X + Y = 81 or X = 81 - Y
The second says 2X - 3Y = 62.
Now you have a system of equations that can be solved by addition or the
substitution method.
We get 2(81-Y) - 3Y = 62
162 - 2Y - 3Y = 62
5Y = -100
Y = 20
If Y = 20 then X = 81 - 20 or 61
Let's check the answer in the first equation: 61 + 20 = 81
Let's check the answer in the second equation: 2(61) - 3(20) =
62
or 122 - 60 = 62
Example:
One group of people purchased 10 hot dogs and 5 soft drinks at a cost of
$12.50. A second bought 7 hot dogs and 4 soft drinks at a cost of $9.00.
What is the cost of a single hot dog? A single soft drink?
Solution:
Declare your variables, and since we are looking for two different costs,
let's call X the cost of a hot dog and call Y the cost of a soft drink.
Now re-read the problem with those variables in mind. We get two
equations:
10X + 5Y = 12.50
7X + 4Y = 9.00
We can now solve by addition or substitution.
We find X = 1 and Y = .50
So the price of a single hot dog is $1, and the price of a single soft
drink is fifty cents