~ What is an Equation?

An equation is a mathematical statement that shows the two sides of the equal sign are the same or equal to one another. For example, the picture on the left is describing the different parts of the equation. The variable which is a number that we don’t know is an X in this equation and there are two values. One of them being the five on the side of the variable and the ten on the other side.  In order to solve this equation, we need to find what the value of our variable is in order to get a solution. In this unit, I’ve learned that you can either take away numbers, and add them. You can even add variables. You can add positive and negative numbers and variables. There are certain rules that we have to follow in order to get the correct solution. We can take away positive numbers if there are positive numbers on the other side by taking away the value. For example, in the picture above, there is a positive five on one side, and a positive ten on the other. In this case and always when doing algebra, we want to find out the value of just one X which means we need to move the numbers on one side, and the variables on the other side. So what we would do now is, add negative 5 to both sides. What this does is it creates a zero pair on the left side and on the right side we have five. This makes our equation a lot smaller and easier to work with. If we rewrite our equation we have X = 5. That’s it, we figured out the value of X. The same thing with negatives. Let’s say this equation was X-5 = -10. It’s the same equation but with negatives, so instead of subtracting five, we want to add five to the sides. Remember, whatever you do on one side, you have to do it to the other side. Before we move on, I didn’t address what a zero pair is. The picture below represents a zero pair. A zero pair is when a positive number and a negative number cancel each other out. For example, the picture shows -1+1= 0. If we have two opposite integers with the same number, they automatically cancel each other out, creating a zero pair. 

~ What is an Equivalent Equation?

  An Equivalent Equation is an Equation that can be equal to an equation. In this picture I drew an equation. The equation originally was -2x+7 = 5x-8. I then proceed to find X. What I did was to get rid of the -2x, I had to add 2x to it creating a zero pair and I added it to the other side as well. The equation becomes, 7 = 7x-8. This is an equivalent equation because I am able to get to my original equation by doing a move. So instead of adding 2x I know subtract 2x. This makes the equation equivalent. Now if we move on, to get the X on one side and the numbers on the other, I added eight to both sides. The equation now becomes 15 = 7x. This equation is equivalent to the first one as well. I’d like to think of equivalent equations like a pattern. If you change the equation it becomes equivalent because you can do a move to get you back to the original equation. For the final step, I proceeded to actually find out what the value of X is so I isolated the X and I got 2 . All of those steps that I did to get X was all equivalent to the original equation. That’s one of the cool things that I learned about linear equations.

~ How to Solve Equations?

Knowing all of these steps how do we solve equations? Is there more than one way? Well let’s get started. We already know how to solve equations algebraically because of the examples I shown above but are there any other ways? The answer is yes. There are two more ways we can solve Linear Equations. The first one is with algebra tiles. I have shown a picture that demonstrates how to draw these equations. This method isn’t very good because it takes a very long time. This method is good for beginners trying to learn Linear equations. In this case, my X variable is the rectangles and the small squares are my numbers. The equation that I made was 3x-1 = -2x+5. I demonstrated it by making the positive numbers or variables coloured in and the negative numbers or variables just white. This is the same idea as the equation solving I showed under the previous columns.  I show what I add or take away by showing the number and then the shape. I started off by taking away the negative 2x so I added it to one side creating a zero pair on the right side, and then I added 2x to the left side. I then rewrote the equation making the new equation, 5x-1 = 5. I wanted to get rid of the negative one so I added one to the left side creating a zero pair and then adding one to the right side. The equation now becomes 5x = 6.For the last step, I divided to find X so I ended up getting 1 .

Another way is a method called BFSD. BFSD stands for brackets, fractions, sort and divide. In the picture here, I chose a question with all the steps that use BFSD. The first move to do in the picture is to get rid of the brackets. In this case, we are using a term called distribution. This means that everything in the brackets are being multiplied by two. Then our equation becomes simplified to X+4 = X+3. For the fraction part, we find the lowest common denominator or LCM for short. Since there is only one fraction, the LCM in this case is three. Now here comes the tricky part. We now multiply everything by three. For the fraction , we multiply the two by three and then it becomes . Remember, a fraction can also be a dividing question so we can just divide the 6 by 3 which gives us 2. Our question looks a lot easier. It went from having brackets to then having fractions, but in just two steps, we were able to make our question look a lot easier than it was before. This also helps us because the question becomes more familiar and easier to solve because having fractions and numbers in an equation can be difficult so just having numbers makes it a lot easier to solve the equation and to find the solution.

~ How can we check to see if our Question is right or not?

In this picture, I made a random question and I solved it. With all the moves and options I knew, I got X is equal to . This is where it gets interesting. In order to find out how to check your question, wherever there is a variable in this case an X, you put your answer there. For example, look at the right side of the picture. I put my answer in brackets where the variables should’ve been. From there you are doing the same thing to solve any Linear Equation except the only difference is there is no variable this time. Getting back to the picture, I multiplied the numbers beside the brackets then I rewrote the equation. I ended up getting all of the numbers to be fractions so this was going to be easy. I found the LCM for all the fractions which was twelve and I multiplied all the numbers by twelve. At the end I got 2+9 = 8+3. Now you may be thinking okay well, what does this have anything to with my last equation on the left side? Well If you add the numbers the final answer is 11=11. This means my answer is correct because both sides equal the same number, therefore my answer is correct. I found this method really cool and very helpful.

~ How was this unit for me?

I thought this unit was going to be very difficult because I didn’t really know what the concept of making things equal on both sides was and that whatever you do to one side, you have to do to the other side was. I was very confused but with a little help, I started to realize that I was overthinking things. I just needed to follow a set of moves. I really liked this unit because every time I was introduced to a new sub topic under Linear Equations, I thought I was going to become confused again but at the end when it was explained to me, it’s like every step of the way made sense. I really enjoyed this unit and i’m surprised by what I’ve learned about Linear Equations and how cool they are!

Scources:

https://study.com/academy/lesson/how-to-write-equations-formulas.html

https://slideplayer.com/slide/8395776/

All other Photos were made by Myself.