What is  a linear relation?

A linear equation is a mathematical equation that can be shown in a graph or in a mathematical equation. Linear means that something is following the expected order or sequence, so a linear equation is an equation that follows the same pattern. if the pattern is linear it has to go up the same amount each time, and if it is not linear it can go up different amounts.

this pattern is an example of a linear pattern because it started at 5 but now increases by 4 each time.

How do you find a rule for a pattern?

For this example you can tell that these are linear because they are going up by 40 each time. on the left side of the T chart there is the title t. on the right side there is the title d. How you could get an answer for how much it goes up by each time would be on the 5 on t and 200 on d, if you divided the 200 by the number across from it you would be left with 40. Now you would have to check with the first number as well and you can tell that 40 divided by 1 would be one so this is a linear relation. the way you would display it would be t = 40 because for every t that there is on the left there will be a 40 on the right. if this pattern is bigger what i would do to confirm that it is right is find  in this case what would be the first 3 on the t side and the last, and then divide them by that number you got. in my case that would be 40 and that is how you can be completely sure that it is right.

How can you determine whether a pattern is increasing or decreasing?

the way to determine if the pattern is increasing or decreasing is by looking at the right side of the t chart and from there you can check whether the number is increasing or decreasing.

at first this looks like a decreasing pattern but when you look closer you realize that it is actually increasing because the 1 is paired with a -2 and the 2 is paired with a 1. from seeing that you can tell that the equation is increasing by 3 for each x it goes up by. the mathematical equation for it would be +3x because for every 1x it goes up the other side would be going up 3.

What is a T-Chart and how do you fill it in?

this t chart is already filled in but if it were blank then you would be given something like this

 

How to plot a point?

what we need to plot the points is the graph to plot them on and the T-chart. Lets say we are plotting the first x first. 1 x and 1 y. the coordinates would be (1,1) for this example it would be easy because both of the numbers are the same but when you look at the coordinates the firs number is located on the x and the second number in the coordinates is the y axis. The pattern shown on this t chart would be x=y and since you know that it makes plotting it a lot easier. this is what x=y would look like.

 

How to graph vertical and horizontal lines?

this one would be vertical. The way you plot a vertical graph would be you find the spot on the x axis you want it to cross which in this case is x=-3. to make a horizontal line it is basically the same except instead of doing x=, it would be y= __.

this is exactly the same as the one before except instead of crossing over the x axis it is crossing over the y line.

Vocabulary:

x and y axes-

the x and y axes are 2 lines that create the coordinate graphs. the y axis runes upwards and downwards, and the x axis runs left and right.

t-chart-

a t-chart is basically a big T. It has 2 titles and then numbers on both sides. normally the left side will have numbers going up starting at 0 or 1 and the right side has numbers that are equivalent like if x=12, then 2 on the left side would be 24 on the right because that’s 2 x 12.

coordinate-

the coordinates are the numbers that are on the x axis and the y axis. coordinates for 1x 3y would be (1,3) because the x axis goes first than the y.

quadrants-

there are 4 different quadrants, the first one is quadrant 1 which is positive x and positive y. next is quadrant 2 which is negative x and positive y. then quadrant 3 which is negative x negative y, lastly quadrant 4 with positive x negative y.

origin-

the origin is the coordinates (0,0) it is the very centre of the graph where the x and y axis connect.

plotting-

plotting is putting dots on the graph according to the coordinates.

linear pattern-

A linear pattern is when the plotted points on the graph make a pattern. Like if the number goes up the same amount each time

increasing pattern-

an increasing pattern is when in each different figure goes up as you go.

decreasing pattern-

a decreasing pattern is when each different figure goes down as you go.

horizontal line-

a horizontal line is a line that goes through the y axis because it only goes sideways.

vertical line-

a vertical line is a line that goes through the x axis because it goes up and down.

Something Else I learned:

that flipping equations to negative makes it come from the opposite direction.

Screenshot of Initials   

What is an inequality?

An inequality is 2 numbers in an equation separated by up to 6 different signs the first sign is less than, the second sign is greater than, the third sign is less than or equal to, the fourth one is greater than or equal to, and the last two are equal to and, not equal to. Lets say that the equation is x < 11. That would mean that x could be anything less than 11. So it could be 10, 9, 8, 7, 6, 5, 4, 3, 2 ,1 ,0, and anything into the negatives as well.

What do the symbols mean?

with greater than and less then ( >, < ) the way that the symbol is pointing always points towards the smaller number. An example for that would be 7 > 6 because 7 is greater than 6. The next is equal to or less than, and equal to or equal than. That works the exact same except for the number could be the same or anything greater or less. If the questions was x is less than or equal to 7 then the answer would be 7 and anything below. The last two are equal to and, not equal to. Equal to is what you would do for the kindergarten classes like 1 + 1 = 2 because it is a true statement. The other one is not equal to which is the equals sign with a line through it and it would be the exact opposite of the normal equals. for example 1 + 1 ≠ 3 because 1 + 1 does not equal 3.

How to solve an inequality?

You solve an inequality basically the exact way that you would have done a linear equation. the number 1 strategy to solving these is using legal moves. that means that whatever you do to one side of the equation, you have to do the exact same thing to the other side. The whole goal is to end up where you have isolated the variable on one side and the number on the other.  An example question will be, 2x -5   1x + 7. What you would do is start off with a legal move like removing 1x from each side so we are left with x – 5   +7. The next step would be to add 5 so it zero pairs the -5. That will leave you with x   12. That means that x is greater than or equal to 12 so the answer could be  12 and up. However, when you are dividing by a negative you need to flip the greater than/less than sign. Let say the question is -3x < 12, after divided you are left with x < -4, but since it is a negative you flip the sign and your answer becomes x > – 4.

 

How do we graph inequalities?

When you  graph inequalities you have to use the greater than and less than  signs as a dot with a hollow inside. The hollow dot represents that the equation is not a less than or equal to, or more than or equal to. Because of that when you are graphing it you draw a blank circle of the number that x is greater than or less than. Then according to which one it is you draw a line towards the lesser side or the greater side. When you do it with greater than or equal to, and greater than or less than you do it the same way except you do a circle that isn’t hollow and the same thing with the line. This way means it could also be equal to.

How to check a solution

There is more than one way to check a solution. The equation i picked to test is 2x -5   1x + 7. to get to the answer the first thing i would do is make zero pairs by removing 1x and adding 5. Now you are left with x   12. the way we check the answer is by putting a number that in this case is greater than or equal to 12. If it is a true statement then you know that the answer is correct. Another way is by reversing what you did and trying to get to the question you started with but you can only use legal moves. If you do only legal moves than it will be right. For example x   12, to get back to what we started at we subtract 5. that makes it x -5   7, next we add 1x and it leaves us with the starting question of 2x -5   1x + 7

https://www.mathsisfun.com/algebra/images/inequality.svg

https://quickmath.com/webMathematica3/quickmath/graphs/inequalities/basic.jsp

 

 

What is an Equation?

An equation is a statement that shows 2 sides of equal equations and are divided by an equals sign. The purpose of these questions is you want to use legal sequences (which is when you add, subtract, divide, or multiply from both sides equally) and you deduct down to where the variable is isolated and one side has the variable and the other has just the numbers.

What are equivalent equations?

Equivalent equations are algebraic equations that has numbers and letters for the variable. When you solve equivalent equations you have to do the exact same thing to both sides until you have isolated the variable. When you solve an equivalent equation every legal move that you do is equivalent to the last one because the only difference between the 2 is adding or subtracting the same number. It can be put back to the original equation using only legal moves

How to solve equations?

What you do to solve the equations is remember that they are split in 2 by the equals sign and then you do legal equations equally to both sides. For example if you added 6 to one side then you would also have to do it to the other side. The whole goal is to this until you have isolated the variable and then you can figure it out using math we have previously learned. Lets say that the question is 4x + 7 = 2x + 12. The first move I would do would be remove 2x and get a zero pair. That would leave me with 2x + 7 = 12, This equation is equivalent with the starting equation because i used legal moves to get to this point. Next up I would remove 7 and then what i am left with is 2x = 5. simplifying it down to x = 2.5 is still an equivalent equation to starting equation.

How to verify.

verifying the number is very similar to how you solve it. You put the number you came up with as your answer in the spot that was the variable before. Then you do the exact same legal addition or subtraction until you get back to the same question you started with. If you cant get to the question you started with using only legal moves you did it wrong but if it the same you have done it right

Vocabulary

Equation- an equation is when there are 2 different sets of numbers and they are divided by an equal sign.

Equivalent- Equivalent means that the 2 things are the same. Equivalent equations which is what we are doing can appear different but if you added for example 4 to each side they would still be equivalent equations because it can be reversed with legal moves.

Solution- A solution is the answer to the question. An example would be 3x = 9 the solution would be 3.

Coefficient- a coefficient is the number before the variable displayed. It lets the person know how many of the variable there are.

Zero pairs- zero pairs are made when you add a legal move to the equation and it makes it zero. An example would be 4x + 2 = 11 how you would make a zero pair by subtracting 2 on both sides. it would become a zero pair because you subtract 2 from a positive 2 leaving you with zero. That would leave it as 4x = 9. Zero pairs are crucial for finding the answers to linear equations.

Variable- a variable is what the x is in linear equations. it can be done with any letter but it is representing the unknown. The whole point of the linear equations questions is to solve how much 1 of the variable is worth.

Constant- The constant is a number in the equation that does not have a variable attached. Constants are worth whatever they say they are even when the answer is completed.

Common Denominator- The common denominator is when you have 2 fractions that have the same denominator. It is important because it makes it so you can easily add subtract or divide fractions.  The way you can make a common denominator when they are different is you have to find the lowest common multiple. Lets say your denominators are a 6 and a 7. The lowest common denominator is 42. Then what you do is multiply the numerator by the amount of times you multiplied the denominator and then they are equal.

Distribute- distributing is when there is a number in front of a bracket so you multiply each thing in the bracket by the number outside of the bracket separately and then the number on the outside disappears.