Week 18 of Math 10 and Week 8 of cautious classes. Never thought I’d have my birthday during a pandemic but here I am, too bad COVID-19 prevented my mom and I from visiting family back in the Philippines. This week had quite a bit of new topics, in this blog post I will be covering a few things I missed from last week like how to determine the number of solutions of two equations alongside some new topics like the elimination method.

There are only three types of solutions when it comes to linear equations, and each of them can be told from a glance. First, there can be no solutions. In this example here, 2x+4 and 2x+1, both do not intersect therefore cannot have a specific point. This occurs if both equations have the same slope, but not the same y-intercept.

Next, which is the one we’ll see most often is 1 solution. This is when the slope is different for both numbers. So 2x+2 and 4x+2 would look like this.

For the last possible solution, or rather solutions, is there can be an infinite number of solutions if both equations are the exact same. If both equations were 5x+7, then both would be in the exact same spot which would mean it would be intersecting itself infinitely.

The next topic we’ve learned is a new way to solve for the solution which is called elimination. This method can be used if there is no variable that is already isolated, for example: 2x+7y=-5 and 5x-7y=19.

The first step to the method is to find a variable that cancels each other out, conveniently 7y is our zero pair meaning we can isolate for x. Next, simply add all numbers straight across then isolate for x.

Now that we know that x=1 we can plug it into one of the two equations to isolate for the other missing value.

Now we know what the solution is with the coordinate (1,-1). You would want to verify if it’s the correct answer however, so plug in the coordinate into both solutions, if it ends up as true for both then it is the correct answer.

Sometimes you will receive a question that does not have a variable that can be easily isolated like 3x+2y=7 and 9x+8y=22. It is possible to solve for this, start off by multiplying all the terms of the first equation by 3 (to isolate for y). You are also allowed to use subtraction or addition depending on the question. For this one subtract both equations by each other and isolate for y.

Then all you have to do is follow the same steps done before. Plug in the value to isolate for the other missing side.

Then verify using the coordinate solution.

And you’re done!

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