Writing Systems of Linear Equations from Word Problems

As we learn about math, we often ask ourselves, "How is any of this knowledge useful in life? When will I ever need to use algebra?" Here''s the thing: When you know how to "translate" word problems into algebraic equations, you''ll immediately see how useful math really is. With a few simple steps, you can turn real questions about the world around you into equations, allowing you to calculate things that you never thought possible. It''s one of the most interesting things about math -- so let''s get started!

How to tell when a word problem can become a linear equation

First, we need to keep our eyes open for a number of clues. These clues tell us that we can turn our word problem into a linear equation:

• There are different quantities of things, such as a specific number of people, objects, hours, and so on.
• Each quantity has a clear value. Instead of saying, "There are a few boxes," the word problem needs to tell us how many boxes there are. Are there five? Six? Seven?
• We need to know at least some of these values to build our linear equation. The unknowns can become variables, like "x" or "y."

How to turn word problems into linear equations

If we follow a few simple steps, we can turn certain word problems into linear equations:

1. Take a second to think about the "problem." What are we trying to find out? What is the "variable" in this word problem? Define all of the words carefully. If we can''t define the words properly, our equation won''t be accurate.
2. Turn the word problem into an equation. Plug in all of your known values and use letters like "x" and "y" to represent the unknown variables. Make sure you write out what each variable represents below the equation so you don''t forget.
3. Solve the equation. Using our math skills, we can now solve the problem and find the values of our variables. We can use a wide range of strategies to solve the equation, including substitution, elimination, or graphing.

An example of a word problem translated into a linear equation

Now let''s see how this all works with an example. Here''s our word problem:

We decided to go to a music concert with all our friends, including 12 children and 3 adults. We paid for everyone''s tickets for a total of $162. Another group of friends paid $122 for 8 children and 3 adults. How much does a child''s ticket cost, and how much does an adult''s ticket cost?

1. Understand the problem

We know two values: 12 children and 3 adults cost $162, while 8 children and 3 adults cost $122.

What we don''t know is how much an adult''s ticket costs, and how much a child''s ticket costs.

Let''s create variables for those unknowns:

x = the cost of one child''s ticket

y = the cost of one adult''s ticket

2. Translate the problem into an equation

We know that 12 children and 3 adults cost $162. Let''s plug in our variables and create an equation based on this:

We can do the same for the other group of concert-goers:

3. Solve the equation, On one weekend they sold a total of 12 adult tickets and 3 child tickets for a total of 162 dollars, and the next weekend they sold 8 adult tickets and 3 child tickets for 122 dollars, find the price for a child''s and adult''s ticket.

When we have two very similar equations like this, we can simply subtract them from each other to get the values we need:



Now that we know the value of x, we can use it to find y.



Now we know that a child''s ticket costs $10, while an adult''s ticket costs $14.
We can now check our work by plugging our solutions back into our original equations:


You can use this strategy to solve all kinds of everyday math problems you encounter in life!

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Get more help with linear equation word problems

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