Being able to solve systems of equations in two variables is great, but it's even better if you can use this skill in solving problems. In this section we'll try out a few examples. Ready to begin?

Example #1

Joe can buy a CD and two books for $28.00, and he can buy two CDs and three books for $48.00. How much does a CD cost?

Solution #1

We begin by identifying the information we don't know, and assigning variables to them. It's easy to see in this problem that there are two unknowns. We don't know how much a CD costs, and we don't know how much a book costs. So why don't we use the variables c and b. Let's write out in words what each variable means:

c = the cost of a CD
b = the cost of a book

Now we look at the information in the problem and try to turn it from word sentences into equations:

(The cost of a CD) plus (the cost of two books) is $28.00
c + 2b = 28

(The cost of two CDs) plus (the cost of three books) is $48.00
2c + 3b = 48

Now we have two equations, and we can solve:

c + 2b = 28
2c + 3b = 48

I will leave the solution to you as a practice exercise. Remember that the question asks for the cost of a CD, so your answer will be the variable c.

Example #2

Jane has 16 coins, all of which are nickels and dimes. The value of all his coins is $1.25. How many nickels and how many dimes does she have?

Solution #2

Again, we identify the things we don't know. We don't know the number of nickels and we don't know the number of dimes. That's two variables (shall we use n and d?), so we expect to be able to write two equations.

n = the number of nickels Jane has
d = the number of dimes Jane has

Since she has sixteen coins, and they're all nickels and dimes, we can write:

(The number of nickels) plus (the number of dimes) is sixteen
n + d = 16

Since the coins have a value of $1.25, we can say:

(The total value of the nickels) plus (the total value of the dimes) is $1.25

Uh oh...we don't have a variable that represents the total value of her nickels, do we? Or the total value of her dimes! But we do know this - each nickel is worth $0.05, and each dime is worth $0.10. The total value of her nickels is the number of nickels times the value of one nickel. Similarly, the total value of her dimes is the number of dimes times the value of one dime:

(The total value of the nickels) plus (the total value of the dimes) is $1.25
0.05n + 0.10d = 1.25

I don't know about you, but I'd rather not deal with those decimals, so would it be okay if we multiplied the entire equation by 100?

100(0.05n + 0.10d) = 100(1.25)
5n + 10d = 125

Ah! Now you've got two nice equations, and you can finish the problem yourself:

n + d = 16
5n + 10d = 125.

Example #3

Greg's age is ten less than twice Verity's age, and Verity is twenty-seven years younger than Greg. How old is Verity?

Solution #3

There are two things we don't know in this problem:

g = Greg's age
v = Verity's age.

Now let's write some equations:

(Greg's age) is ten less than twice (Verity's age)
g = 2v - 10

(Verity's age) is 27 less than (Greg's age)
v = g - 27

In this problem we have two equations that are both solved for a variable; it looks like subsitution might be the easiest way to complete this one. Go ahead and work it out as a practice exercise!

Example #4

The sum of two numbers is three more than twice the smaller number, and the difference between the numbers is 3. Find both numbers.

Solution #4

For this problem we have a smaller number and a larger number that we don't know:

S = the smaller number
L = the larger number

Why did I use capital letter variables this time? Because I know that if I use a lower case L, I'll end up mistaking it for a 1 and get myself in trouble. Now we create our equations:

(The sum of two numbers) is three more than (twice the smaller number)
S + L = 2S + 3

(The difference of the two numbers) is three.
L - S = 3 (note that we subtract the smaller from the larger, since the difference is given as a positive number).

We have two equations. You can either rearrange to do substitution, or you can arrange to do addition. The choice is yours!

1.

Finish solving example #1.

2.

Finish solving example #2.

3.

Finish solving example #3.

4.

Finish solving example #4.

5.

Ferdie has 27 coins, all nickels and pennies. The value of her coins is $0.75. How many of each does she have?

6.

The amount of snow that fell on Thursday is two inches less than twice Saturday's snowfall. Saturday's snowfall was 10 inches more than Thursday's. What was the total amount of snow for the two days?

7.

Mara is seven years older than Sarah. Together, their ages add to 61 years. How old is Mara?

8.

A candy bar and a soda cost $1.95 together. The cost of two candy bars and three sodas is $5.00. How much does each item cost?

9.

My children's ages (in months) add to 79. One child is 29 months older than the other. How old is the younger child?

10.

If Trevor mowed 5 lawns and weeded 6 gardens, he would have earned $398. If he had mowed 2 lawns and weeded 2 gardens, he would have earned $140. How much does he charge to weed a garden?