Writing Equations for Word Problems

Sometimes figuring out how to write an equation for solving a word problems is the hardest part of the solution.

Here are some examples of word problems and the equations they lead to:

The sum of two numbers is 90. The difference between them is 75.
Find the two numbers.

Let a and b stand for the two numbers.

What do you know?

You are adding and subtracting the same two numbers.

Can you write one equations using addition and one using subtraction?

(Sum) = a + b = 90

(Difference) = a - b = 75

You could solve these equations using substitution or elimination.



A wildlife management station cares for sick birds and deer. The animals being cared for have a total of 25 heads and 74 feet (no heads or feet are missing!). How many of each animal are there?

Let b = birds and d = deer

What do we know?

If there are 25 heads, there must be a total of 25 animals

b + d = 25


What else do we know? Birds have two feet each and deer have four feet each and all the feet total 74.

2b + 4d = 74

Now we could solve by either substitution or elimination.


Breakfast: If two eggs with bacon cost £2.70 and one egg with bacon costs £1.80, what does bacon cost alone?

What do we know?

If we use e for eggs and b for bacon, we can write equations for each meal:

two eggs and bacon costs 2.70

2e + b = 2.7

one egg with bacon costs 1.80

e + b = 1.8

We can use substitution or elimination to solve these equation.

Walter is riding his bike to Thames City which is 28 miles away. It takes him one hour to make the ride against a head wind but it only takes him 48 minutes to return to St. John's Wood with a tail wind.

How fast was he riding without the wind and how strong was the wind?

If a is his average speed and w is the speed of the wind..... what else do we know?

(a - w) would be his speed (rate) riding against the wind.

(a + w) would be his speed (rate) riding with the wind

The distance for both parts of the trip is 28 miles.

It took one hour on the ride against the wind and 48 minutes (48/60 = 4/5 hour) to ride with the wind.

Using d = rt

28 = (a - w)1 (against the wind)

28 = (a + w)(4/5) (with the wind)

Simplify:

(1)28 = (a - w)1

28 = a - w

(5/4)28 = (a + w)(4/5)(5/4)

75 = a + w

We could then use substitution or elimination to solve the problem.

Stephanie received the results of her ERB scores in math and reading. Her reading score is 70 points less than her math score. Her total for the two parts is 1250.

Let m = math score and r = reading score.

What do we know?

Her math score plus her reading score is 1250.

m + r = 1250

What else do we know?

Her reading score is 70 points less than her math score.

r = m - 70

We could use substitution or elimination to solve this problem.