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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Week 17 of Math 10 and Week 7  of cautious online classes. We are now learning a new topic which is the called the System of Linear Equations. In this blog post I will be covering what it is and how to solve for it without the use of graphs.

A system is essentially a set of linear equations that work together, and is described through two equations that can be written into lines. The point at which the two lines cross is called the solution of the system, which is a specific coordinate and the only coordinate where both lines intersect. For example, if we had 2x+y=4 and 3x+y=6 as our two equations, the solution would be the coordinate (2,0).

In order for us to confirm that this indeed is the solution we have to plug the numbers from the coordinate into their respective variable. If we do that we get 2(2)+0=4 which is true, and 3(2)+0=6 which is also true. If both equations result in a “true” statement then the solution is correct.

However, we while it was easy for us to insert the number into a graphing calculator like Desmos, we should also be able to know how to  solve for the solution algebraically. So let’s try to solve a new question without the use of a graph. This time our two equations are 5x-3y=27 and 2x+y=2. First, we start off by choosing an equation, then isolating either x or y. We can start off by doing equation 2 with the intent of isolating y.

Now that we know the value of y, we can plug in the y value into equation 2.

Now that we know what the x value is, we can go backwards and plug it into the previous equation to find the actual value of y.

So now we know that x=3 and y=-4, therefore the solution is (3,-4). Finally, you would want to double check if the solution is actually correct by inserting them into the initial first two equations to see if they’re true.

And you’re done!

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Week 16 of Math 10 and Week 5 of quarantined classes. For the first time in a while we’re actually able to go back to school, but under heavy care as to prevent the spread of COVID-19. Online classes are still an option though so I’m gonna play it safe and stay at home, I’ve still managed to learn a lot despite the restrictions of video calls. In this blog post I will be covering Slope y-intercept form, Point-Slope form, General form, and how to convert them into each other.

There are three different ways to write a linear equation:

Slope y-intercept form: y = mx + b

Point slope form: m(x-x) = y – y

General form: Ax + By + C = 0

If we’re given the values m=2 with the coordinates (3,6) we should have all we need in order to write in all forms. Let’s start off with point-slope form. Point slope form has the slope as the value behind the parenthesis while the coordinate values go into the second x and y of the equation. Following the previously given values the equation would look like this.

If the coordinate was negative, we simply change the minus signs to positive because two minuses equal a positive.

For general form, all values have to be on one side while zero is on another. With the above values we can’t immediately go to general form however, because we need to convert it into point slope, then into slope intercept form to convert it into general form. To turn it from point slope to slope intercept, first distribute the slope into the parenthesis

Then try to isolate the y in the equation

Afterwards, simply move the y into the other side and replace it with a 0.

That’s it, usually the most important form you have to go from is point slope because it can convert into general form and slope intercept. As a bonus I’ll be covering how to change general form into slope intercept.

For our general form question we’ll start with 4x-2y+10=0, first step is to move the y into the other side

Next, turn the y into 1y through any means. In this case divide y by 2, then divide all terms on the other side by 2 as well. You now have it in slope intercept form.

And you’re done!

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Week 15 of Math 10 and Week 4 of quarantined classes. This week has been fairly quiet, nothing absolutely massive or no new revelations, which is nice considering how this quarantine makes lessons harder. In this blog post I will be covering Parallel and Perpendicular lines, a topic that I was unable to add into my previous blog post, alongside Linear functions.

Parallel and Perpendicular lines are types of lines that are adjacent or directly on a certain line. Parallel lines are lines that will never intersect with the main line.

In this image, the blue line is our main line while the green line is it’s parallel. Notice how both lines have the exact same slope but are situated on different parts of the y-axis. The slope of the first line is always equal to that of the second line if it’s a pair of parallel lines.

Perpendicular lines on the other hand are the complete opposite. These lines will always intersect with the main line no matter what.

Here, you can see that the red line intersects with the main blue line,. The slope of the main line is 2 or while the slope of the red line is -0.5 or . If you had a keen eye you’d notice that the slopes for both lines are polar opposites, or in math terms, negative reciprocals of each other. So in order to find the slope of a perpendicular line you have to flip the slope of the main line on it’s head then turn it into a negative. Another note is that since these lines are negative reciprocals, they will always result in -1. multiplied by is equal to or -1.

So if you were given two different slopes e.g and 2x, this is how you’d solve for both parallel and perpendicular lines.

If you’re looking for a parallel line, you know that the slope of the first line is equal to the slope of the second line, or = . Your goal is to isolate x, so plug in the values then follow the order of operations.

If you’re looking for a perpendicular line, then the formula for finding it is because perpendicular lines are always =-1. So plug in the values then follow the order of operations like above.

Next in line are Parent Functions. Parent functions are lines that are the base version of all functions. They have a slope of 1, have both an x and y intercept of (0,0) and are written as y=x. This is how they look like on a graph.

If other numbers are added to the Parent function they become something else entirely. In this case they can become linear functions which are written using the slope y-intercept form, or y=mx+b. M refers to the slope while B refers to the y-intercept. When it comes to solving equations the value you’re looking for may vary, but its all a matter of what information is given to you. Here, I’m going to provide three different questions and the methods to approach them.

M = -2, passing through (3,6). For this question we already have the slope given to us but we don’t know how the values for x, b and y. Luckily, since coordinates are formatted as (x,y) we know that the values for x and y are 3 and 6 respectively. Plug in the numbers then follow the order of operations.

The answer comes to y=-2x+12.

Goes through (8, 6) (4, 3). For this question, we aren’t given anything except two coordinates, but while this may seem impossible to do, remember to always try to find the slope first. In order to find the slope, use the slope formula .

Now that we know the slope is we can make the slope interval equation y=x+b. The next step is to choose a coordinate to plug in the values, then follow algebra. It actually doesn’t matter which coordinate you choose, both will share the same answer.

Since B=0 the answer just ends up being  y=.

Perpendicular to y=+4 going through (8,2). Now perpendicular and parallel are coming into play. Since the question says perpendicular, we have to flip to become or 4, but there are two different y-intercepts that are shown. The +4 in the slope interval and the 2 in the coordinate, so which one do we plug into the equation? The answer is always the coordinate, meaning that even if the coordinate were replaced with “y-intercept of 2” you still use 2 because a y-intercept of 2 is just (0,2). Plug in the numbers then algebra takes over.

And you’re done!

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Week 14 of Math 10 and Week 3 of quarantined classes. We haven’t shifted much from the properties of graphing, but we did learn a new topic which is Slope. In this blog post I will be explaining what slope is and how to solve for it.

Slope is a value that shows how steep or shallow a line is on a graph. This is the also the same as the tangent ratio because tangent’s opposite and adjacent sides can be substituted for the X and Y values on a graph. It’s commonly represented through , so if our line were to look like this

Then we just have to count how much the x value rises and how much the y value runs left or right. This is also known as the method. In this case right here our coordinates are (0,6) and (-3,0) so we start from the lowest point, go up or rise until we reach the second point, then run to whichever direction the second point is. The end result is or 2.

Another method to finding the slope is by finding all the “nice points” or the points of whole numbers where the line intersects, then find the within those points.

This is useful if you’re not given two specific points or if you want to try your hand at finding the slope faster than the first method. There are four different kinds of lines and slopes that you should also be aware of: Positive, Negative, Zero and Undefined.

The first two (Black and Green) are self-explanatory, they’re your typical sloped lines that face left or ride according to their x-value being negative or positive. The other two are a bit more complicated. Zero-sloped lines (Blue) are lines that have their y-value at 0, meaning there is no slope. Undefined lines (Red) are named as such because the x-value is zero, so the slope is essentially meaning that you’re trying to divide by zero (which isn’t possible, hence the undefined).

Next, I’m going to demonstrate how to find the slope using slope formula. Slope formula is and is used if given two coordinates. So if we were asked to find the slope between (6,4) and (3,2) it would look like this

Always remember to start with the y-values as the numerator, highlight it if you can. I’ve personally gotten a lot of questions wrong because I instinctively read the first number of the coordinate -or x- and put it on top which gave me the wrong answer. Another type of question you can encounter is if you’re given two coordinates with a missing value, like here

Here, it isn’t very different from the first method. Follow the slope formula then try to isolate the variable as if it was a person with COVID-19 who left quarantine. If you follow the order of operations you’ll likely get the right answer, just remember if you’re going to do something to one side, also do it to the other.

And that’s Slope!

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Week 13 of Math 10 and Week 2 of online classes. This week we started focusing more on functions and how to express them on a graph and how to interpret them. In this blog post I will be explaining the different ways functions can be shown and how to interpret a graph.

For starters, we can represent functions in three different ways:

1. Equations
2. Mapping Notation
3. Function Notation

Equations are your typical y=x expression. For example, y=3x+4 is a form of a function because it has a degree of 1 making it a linear expression and it does not result in a vertical line which is not a function.

Mapping notation is represented through f:x —–> 3x+4 and this is used as a way to find a specific point rather than give a line. You can find the specific point by plugging in a number into the x value and simplifying. So if f:(-2) then 3(-2)+4=-2 which gives you the specific point and ordered pair of the function as seen here.

Function notation is represented through f(x)=3x+4. Solving this is no different from solving mapping notation, but the difference is that f represents the name of the function, meaning that we can have different functions add or subtract into one another. So if f(x)= 3x+4 and g(x)= 4x-2 and x= 5 we can subtract them.

Next is how to interpret graphs with them. Let’s say we were given a graph that looks like this:

We can find out what the specific point that a function notation gives. If we were given f(7) we can find the point where f(7) and the graph intersect and find the y value, which for this case is 3.

This also applies to finding the x value too. If f(x)=6 then we can estimate that the x value is 4.5 based off the point that 6 intersects with on the graph.

And that’s how function notation works!

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Week 12 of Math 10, and Week 2 of online classes. We’ve gone further into relations through domain and range then moved on to functions. However, domain and range was one of the more difficult topics that I have dealt through, so despite my doubts, I’m going to take it upon myself to try and explain it to make sure I remember what I learned. In this blog post I will be detailing what domain and range is and how to interpret it into set notation.

So, first let’s explain what they are. Domain is the set of all possible numbers on the x-axis, while Range is the set of all possible numbers on the y-axis. You can find the domain and range through a variety of ways such as ordered pairs and graphs, like this one:

Here you can see two pairs of open dots, two on the x-axis and two on the y-axis. To make things easier we will deal with the domain first. Start off by looking for the smallest value that has a dot, which in this case is -4, then look for the biggest value that has a dot which is 6. We format domain and range through set notation which looks like this:

Set notation starts off with the smallest value on the left most side, followed by a less than sign and a variable (usually is x but can vary from question to question) and that variable is also followed by another less than sign and the biggest value. Usually questions that have two points follow this format.

To make things easier, think of x as the stretch between -4 and 6, x has to be bigger than -4, but can only be smaller than 6.

There can also be graphs that have lines with arrows in them, like this one.

These ones are a bit more complicated, as the arrow indicate that the line will continue to go on infinitely. Luckily there is a way to express that in set notation as well. For the domain the smallest value is 2, but since the x is in front of 2 and there’s a shaded dot, x has to be greater than or equal to 2.

Normally {} would be the final result, but because there’s an arrow, x can go on forever and it would be unreasonable to attempt to label every single possible number, we will instead use x∈R or x is an element of real numbers, so the domain would be {, x∈R } For the range, there aren’t any two points we can calculate, so the range is simply 0.

Finally, there’s a graph that has two arrows

This graph will be using rules from the previous questions. Here -5 is the smallest value on the x-axis, and since there are arrows infinitely going to the right, the answer will be {, x∈R}. For the range however, the lines with arrows have a slight curve to them, meaning they will infinitely go up for the upper line and down for the lower line. So it would mean that all numbers along the y-axis are possible numbers. So in order to simplify things, y∈R or y is an element of real numbers will be used for the range because there is no set point for the y-axis.

And that’s how Domain and Range works

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Week 11 of Math 10 has us moving on from factoring trinomials to relations in mathematics. The second week of quarantine learning has been great so far, as I’ve been absorbing and understanding the topics well despite the limitations of online teaching. This blog post will be covering relations and how to express them in various forms.

A relation is a set of data that – as the name suggests – are related to each other. Some examples of relations include:

• The value of a vintage wine and its age
• The battery charge of a phone compared to how long its been plugged in
• Time to drive to a supermarket and distance

Relations are are comprised of two variables which are the independent variable and dependent variable. Although these can be renamed, these two values are commonly referred to as ordered pairs which is an X and Y value inside a parenthesis like (x , y). The independent variable which is X does not rely upon other values, whereas the dependent variable relies upon the independent variable to find the answer.

Relations can be represented in a variety of ways like with diagrams, words, sets of ordered pairs, but for this blog post I will be covering how to solve for x through equations and map them with a table of values

Our relation will be y=6x-2 and we’ll first start off by making a table of values to add them into.

To keep it simple, we will be using numbers 1-8 as our input values, in order to find the value of Y we have to solve the right side. So replace x with 1 and solve accordingly.

Now simply input the numbers into the X to solve for the output of Y.

Now that there’s a full table, you can actually turn these into ordered pairs, which can then be used for other diagrams like graphs.

And that’s how you solve relations.

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Week 10 marks the first blog post while during the COVID-19 Pandemic quarantine. Most, if not all of our lessons are going to be in online classes within the confines of our homes. Personally, I find it hard adjusting to this new way of schooling, but I’ll try my best to both learn and display the knowledge gained from these classes. This blog will be covering what I learned while inside my bedroom as opposed to the classroom.

We reviewed the last topic that we were learning before spring break, which was factoring trinomials. Generally there are three steps to go through when assessing a trinomial question, and it goes like this:

Step 1: Check if there is something common within all the terms, or look for the Greatest Common Factor. If there is a common factor, divide all terms and place the GCF behind a parenthesis.

Note that ALL terms need to have a common factor in order for this step to apply, if even one term is out of line, disregard the step and go straight to the next step.

Step 2: Check if there are two terms. If there are two terms, it’s likely that it’s a Difference of Squares question. A difference of squares question is when the constants are perfect squares and the variables are even, and so we can factor by square rooting.

Two key things to note is that Difference of Squares questions –like the name suggests– need to have subtraction as its operation, if it also has a variable power that is not even or a constant/coefficient that isn’t a perfect square, then the whole step is to be disregarded and we move on to the next step.

Step 3: Check if it follows the formula: . This pattern means that its possible to turn the trinomial into two simple binomials by finding two numbers that can add into the second term while multiplying into the third term.

Sometimes however, you would go through all the steps and find that the trinomial simply can’t be factored and say its not factorable. But although some trinomials may look VERY unappealing, there is a fourth step you can do to try and factor “ugly” trinomials. One such example of an “ugly” trinomial is  which I will show how to factor.

Start off by drawing a 2×2 grid, then place the first term on the upper-left box and the third term on the lower-left box

Now, multiply both and find a factor that adds into the second term and place them into the remaining boxes.

With the grid complete you can now find the final answer. One way to find the factored answer is by looking for numbers that are in common with each other, here and 15x both have 5x in common, so you should align them together like so

In this case, in order to get  you need another x to multiply into it, and in order to get 15x you need to multiply 5x by 3, so align them by their respective boxes like so

Now there’s only one term left, and 2 is the number that can multiply into 2x and 6 from the x and 3 from earlier.Getting the one term usually helps you find the remaining terms. Our final answer is (5x+2)(x+3).

And that’s how you factor trinomials.

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Week 7, the final week before Spring Break and the inevitable COVID-19 quarantine. One of the new topics we learned is how to factor trinomials, previously we were expanding upon polynomial questions but now we’re working backwards. In this blog post I’ll be covering a few different ways to factor trinomials.

The first way to solve trinomials  can only apply to simple trinomials. Simple trinomials are characterized by having a degree of two, no leading coefficient, and have three terms. One example of a simple trinomial is . To find the factored version of this trinomial, you first have to look for the constant which in this case is 15. Next, you have to factor 15 in all the ways you can, such as here.

Then you have to find the factorizations that add up to the middle term’s – or x’s- coefficient which is 7. Here, only 4+3 can add up to 7 while multiplying into 12. Finally, turn the factored numbers into a simple binomial. Our final answer would be (x+3)(x+4)

That’s one way to factor trinomials, but the downside to this is that it can only apply to simple trinomials, so the next method is a more all-encompassing method to factoring any kind of trinomial.

The next trinomial example will be . The first step to solving this is to find the GCF or Greatest Common Factor of all the coefficients. In this case, 6 is the GCF as it divides into 12 and 18. However, it doesn’t just stop here; not only is 6 a common factor, x is also common among all the terms, making 6x our GCF. The next step is to divide the GCF by all the terms, like this:

If possible you can simplify even further, the factored trinomial just so happens to also be a simple trinomial, so we can keep going. The only numbers that can multiply into 2 while adding up to 3 is 2 x 1, so our final factored trinomial would be 6x (x+2)(x+1).

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