Unit 21: Systems Word Problems

** Important Note:  Before completing this unit, you must first be thoroughly comfortable with how to solve systems of equations, systems of inequalities, and linear word problems.  If you are not completely comfortable you need to go back to those units, and review until you are.  This unit is a more advanced topic that requires mastery of the basics first. **

How to identify a system of equations word problem

Look for two things being compared in two ways. Some examples may help:

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended?

This is comparing adult tickets and child tickets in two ways: the amount of tickets and the money they generated.

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.

This is comparing the digits of a two-digit number in two ways: the sum of them and what occurs when they’re reversed.

A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree?

This is comparing two orders of bushes and trees in two ways: how much one order cost and how much a second order cost.

Strategy:

  1. Identify the two things being compared and assign variable letters to them (Use letters that can easily be kept straight: If comparing apples and bananas, use and b).
  2. Set up your first and second equations.
  3. Use Desmos to solve.
Examples

Let’s look at some examples.

At a movie theater, a popcorn costs $2 more than a soda.  If buying 2 popcorns and 4 sodas costs $28, how much do a popcorn and a soda together cost?  

A) $4
B) $6
C) $10
D) $13

Looking at the first sentence, we could create an equation like

$$p=s+2;;;;;;or;;;;;;;;p-2=s$$

We use p instead of here to make it very clear what the variable represents, popcorn. Also note that these are exactly the same equation, we can write it either way depending on what variable we want to isolate.

Looking at the second sentence, we can generate another equation.

$$2p + 4s = 28$$

Because we already have a variable isolated in the first equation, substitution will probably be much easier than elimination.  Here we’ll substitute for p.

$$2(s+2)+4s=28\2s+4+4s=28\6s=24\s=4$$

Now that we have s, we can plug it back into our original equation to find out that p is 6.  Look closely at our answers!  We have both 4 and 6 as answer choices, representing the price of soda and popcorn, but that is NOT what the question is asking for.  We have to find the two of them added together to get answer C).

TIP:  Always check closely that you are answering what the question is actually asking for. The SAT loves to put the values for x and y in the answer choices, and then ask you what some combination of them is.  

In this next example, the answer choices are pairs of equations, so you can form your own equations, then look for the ones that match yours:

Mike’s Meatball Market sold 2,321 meatballs from Friday to Sunday. All the meatballs sold at Mike’s are either beef or turkey. Beef meatballs sell for $1.50 each, turkey meatballs sell for $2 each, and the total sales from beef and turkey meatballs was $3,876. If the number of beef meatballs sold is represented by b, and the number of turkey meatballs sold is represented by t, then which of the following systems of equations, if solved, would determine the number of each type of meatball sold? 

A) (1.5b+2t=3,876\b+t=2,321)
B) (1.5b+2t=2,321\b+t=3,876)
C) (2b+1.5t=3,876\b–t=2,321)
D) (2b+1.5t=3,876\b+t=2,321)

First we should decide if it needs to be t or – t,  and then if it should equal 2,321 or 3,876.  We go back to the word problem, and see “Mike’s Meatball Market sold 2,321 meatballs from Friday to Sunday. All the meatballs sold at Mike’s are either beef or turkey.”  We can paraphrase this as “they sold 2,321 beef and turkey meatballs combined”, or t = 2,321.  We can immediately rule out answers B) and C).

Now we look at the difference between the first equation in A) and D), and see that the b coefficients are different.  Should it be 1.5 or 2b ?  Going back to the words, we see “Beef meatballs sell for $1.50 each”.  $1.50 times the number of beef meatballs is the same as 1.5b , so we now know the correct answer is A).