When you get to grade 10, you’ll start having to find the POI or point of intersection of linear systems. In case you forgot, a linear system refers to 2 linear lines on a cartesian plane. Moreover, to solve a linear system, you will be required to find the POI of those 2 lines. Generally, you will learn 3 common methods to do so: 1) Elimination, 2) Substitution, and 3) Graphing. These methods are useful, but they can require a lot of time to get to the answer (especially with graphing). Fortunately, there is a way that we can derive a faster formula. At first, let’s take a look at this question (below) to see if we can draw some connections between our problem types:

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If the yellow car has a speed of 75 km/h, the blue car has a speed of 50 km/h, and the yellow car is 100km behind the blue car, how long will it take the yellow car to catch up with the blue car?

To solve this problem, we need to look at how much more distance the yellow car is covering compared to the blue car every hour. The reason being, we want to find how much the gap is decreasing by every hour, which will allow us to solve for the total time. Hence, let’s start with subtracting the speed of the blue car from the yellow car:

Alas, the yellow car is covering 25 km of the gap every hour. Now, if the original gap was 100 km, we can easily find the time it will take for the yellow car to catch up with the blue one:

Thus, it would take the yellow car 4 hours to catch up to the blue car. Now, let’s summarize what we did. We got the difference in position and divided that by the extra distance covered by the yellow car. Moving on, let’s see if we can apply this notion to a system of linear equations:

If we look into what the ‘b’ value represents, it is actually the y-intercept. Now, if we were to convert a cartesian plane (or graph) into the perspective of a road (like the problem above), then the y-intercept could represent the position of each ‘car’. Furthermore, the slope values, or ‘m’, could represent the speed of the cars. Finally, if we were to put that into a formula we would get:

You may be confused about why we have 1-2 for position, but 2-1 for speed. The reason is, we are assuming that the second car (like the yellow car) has a faster speed, and we are assuming that the first car is farther in position. Overall, this is how I was able to get to this formula. Finally, before we test it out, let’s derive it from our original system and see how it matches:

As you can see, we ended up getting the same formula. However, I find that the original way is more intuitive as you get a better understanding of why the formula is working. Now, last but not least, let’s test this formula with some problems:

Now, let’s plug in the values into our formula and see how it works:

Let’s check our answers using an online graphing tool (desmos):

The POI is indeed (2,3)!
The POI is indeed (4, 8.5)!

Before I forget, please check out my review of solving equations, if you would like to get a refresher of how to solve equations… Overall, we started with a relative speed problem, thought about it logically, turned it into a formula for a linear system, derived that same formula, and applied it to real problems! This strategy may or may not be allowed on a test, but it’s definitely a great way to check your answers. For emphasis, if the two equations are in slope y-intercept form, you can find the POI in under 20 seconds!

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