Write and Solve Systems of Equations from Word Problems

Many applications in real-life contain two or more unknowns. To solve such problems, we must write as many independent equations as there are unknowns. Otherwise, it is not possible to obtain numerical answers.

This section mirrors what you’ve already been doing this unit – that is, solving systems with two or more variables – except now you’re making the equations (the hard part). Because you already know by now how solve systems of equations, for several of these problems, you’ll only be expected to setup the equations.

Generally, to change sentences into mathematical expressions and equations, look for key words like these:

Increased: +

Sum: +

More: +

Decreased: –

Difference: –

Less: –

Twice: ×2

Doubled: ×2

Tripled: ×3

The same: =

Let’s begin with some basic examples where we learn to go from words to numbers and variables.

Question:   The sum of two numbers is twenty-one and their difference is fifteen. What are the numbers?

Solution:   You being by assigning a variable for each unknown number. Let the first and second number be x and y, respectively. You’re told that their sum is 21 and difference is 15:

  • x + y = 21
  • x – y = 15

From here, you can use the method of elimination or substitution. For instance, using substitution, you can solve for x in the first equation, then substitute the expression into the x of the second equation. Eventually, you should end up with a point of intersection at (18, 3).

Question:   Number A divided by number B equals one-fourth. Number B divided by number A equals two times number A. What are the two numbers?


Start by setting up the equations:


After solving, you should get: A = 2 and B = 8.


Question:   The sum of two numbers is 67. The quotient of the greater divided by the smaller  number is 7 with remainder 3. Determine the two numbers.


Start by setting up the equations; x represents the smaller number, y represents the larger.


After solving, you should end up with x = 59 and y = 8.


Money Applications

Given the complexity of certain finance-related formulas, oftentimes they’ll already be provided to you.

Uniform Motion

For these types of problems, you’ll be required to remember the formula for speed: distance over time.

  • A similar example can be found here.
  • Try this extra challenging example:
    • Runner Y leaves the starting line at 12:00. Five minutes later runner X leaves from the same place and catches up to runner Y at 12:07. Four minutes later, runner X is one hundred meters ahead of runner Y. What is the speed of each runner in m/s?
      Solution →

Applications involving Mixtures

Here’s the basic idea involving mixtures, for a mixture of several ingredients A, B, C, … etc., the total amount of mixture is equal to the individual volumes making up that mixture. For example, A + B = 10 liters. If you’re looking for the volume of A and B, and you know the concentration of an ingredient found in both A and B, you can represent the concentration of that particular ingredient as a decimal factor multiplied to the unknown volume variable. For instance, if A and B represent fuel mixtures (ethanol + gas), the percentage of ethanol in mixture A is 15%, the ethanol in B is 8%, and the final mixture has 11% ethanol, this can be written as: 0.15A + 0.08B = 0.11(10 liters).

Applications involving Work, Fluid Flow, and Energy Flow

In order to create these equations, you need to know the following:

Work:    amount of work done = rate of work × time worked
Fluid flow:    amount of flow = flow rate × duration of flow
Energy flow:    amount of energy transmitted = rate of energy time × time

Notice that in all three scenarios, the given rate is multiplied by an unknown variable (italicized). A work-related application is demonstrated below.


And finally, we revisit the application problem that was introduced at the beginning of this unit:

Here you’re expected to find the current represented by I1 and I2 by constructing equations using Kirchhoff’s law and Ohm’s law. Of course, if you’re not enrolled in an engineering program that involves electricity, you can skip this one.