Sometimes a system isn’t given to you as two separate equations, but as a word problem with separate sentences that look for it’s own values to also be answered in a sentence, here’s how to translate two different types of word problems into a system:

The sum of two numbers is 15. Two times the larger number is 6 less than 3 times the smaller number, find the numbers:

This kind of word problem is one of the simplest and helps to explain how to look for all the necessary information in word problems, the first step is to declare what our variables will mean, in a simple problem like this, x will equal the first larger number and y will equal the second smaller number, next, take note of where verbs are in both the sentences, in this problem, both sentences use “is”, meaning that this is where our equals signs will be in both equations, for our first problem, “the sum of two numbers” means one number being added by the other number, these numbers being our variables means that this simply means x + y and the other side of the equals sign just simply says 15 so we write that down too. For the second equation, our “larger number” is the x variable so we put a coefficient of 2 with it to make “Two times the larger number” or 2x, now, seeing that this sentence has a “less than” in it means that the order we have these in switch, this and knowing that “less than” also means to subtract tells us that this side of the equation would be 3x – 6, and that makes “Two times the larger number is 6 less than 3 times the smaller number” actually mean 2x = 3x -6.

 

A cycle road consists of a series of uphill and downhill sections. The cyclist averaged 20 km/hr on the uphill sections and 40 km/hr on the downhill sections. If they completed the 35 km course in 1.3 hours, determine the length of the downhill sections:

A word problem like this where we calculate distance, speed or time is one where this diagram comes in handy,

Because distance = speed over time, in a case where you for example had your distance and speed but had to figure out your time, see on the diagram that if you ignore the time variable, it shows a fraction of distance over speed, meaning that you would divide these two to get your time, this will be useful later.

When starting this question we first declare our variables by reading the last sentence of the word problem, then we can organize the information we currently have for our distance, speed and time variables by drawing a 3×3 grid, one column for distance, speed and time, and rows for our downhill, uphill and totals of everything which for this problem, would look like this:

Now that we have a way to organize our information, we can fill in the boxes using our clues from the word problem.

We can see that we have a lot of information still but not enough to form a full equation, what we do here is recognize that since the length (or distance) of the downhill sections are what were solving for, the distance column for both the downhill and uphill sections are where we put our variables, and referring back to our pyramid from earlier, since we have distance and speed and need to figure out the time, we divide our distance by our speed which makes our grid now look like this:

Now we are missing one box of our grid but that won’t matter, because we have two columns with totals and x and y being together, we can just use them as both our equations, making this our final system: