This week in Math 10, we learned System of Linear Equations (the point at which two lines on a graph meet). There are two ways to solve systems, one of them is substitution (an algerbraic method of finding solutions for sustems) and the other is elimination (elimination is used when all variables have a coefficient that isn’t one)

Substitution

step 1- when you are given 2 equations, isolate x or y in one of the equations. When you get a new equation, plug it into the second equation

Step 2- solve

step 3- once you find one of the variables, use it to plug it into the other equation you were given. Solve again so you find the other variable.

Step 4- verify that your coordinates are correct by plugging them into the equations given in the beginning to see if they give the same number that it equals

Now we know where the linear equations meet on the graph.

Elimination

Step 1- find a zero pair in the two equations that you were given. Take out the zero pair from the equation and add or subtract the new equations together to form one equation.

Step 2- solve to isolate x or y

Step 3- use the variable that you found and plug it into one of the original equations from the beginning and solve for the opposite variable

Step 4- verify by plugging the numbers into the original equations to see if they add properly.

Now we know where the two linear equations meet

This week in math 10 we learned different forms of equations based on slope. One equation was the slope y-intercept ( y = mx + b, where m is the slope of the line and b is the y-intercept), another equation was general form (Ax + By + C = 0 where this equation has no relation to y intercepts or slopes. Not the most useful equation), the last equation is the slope-point form y-y₁=m(x-x₁) where m is the slope and y1 and x1 are one coordinate that the line goes through. The easiest and most useful equation. These equations are all easy to change into each other.

Slope-Point Form to Slope Y-Intercept Form to General Form

If we are given coordinates (-4,2) and the slope -1/3

First we figure out which equation to use and I chose slope=point form because we are given the variables needed to complete the formula.

m(x-x1)=y-y1

-1/3(x+4)=y+2

Now we distribute

-1/3x – 4/3 = y+2

Thenwe just solve the most we can

-1/3x – 10/3 =y

Now we already have it in slope y-intercept

Then to general form we need to multiply everything by the denominator which is 3 for both of them

-x -2 = 3y

Now we need y to equal a 0 so we could subtract 3y from both sides but we want x to be positive so I am going to take -x and -2 and switch them to the other side of the equal sign

x+3y+2=0

That’s in general form

This week in Math 10 we started a new chapter called linear relations ( any change in an independent variable will always make a change that is in agreement to the dependent variable.)  which is pretty much what we just learned, but a relationship that is the same proportion so when it’s placed on the graph it creates a line.

To find the slope, you must do rise divided by run 

Step 1: Find the rise. The rise is how much the slopes goes up . It corresponds with the y value. Count the difference between the y values.                                                                     In my example, the y values were -3 and 8. The difference is 11 and since the rise is going up, that means its positive.

Step 2: Find the run. The run is how much the slope moves to the sides. It corresponds with the x values. Count the difference between them.                                                                    In my example, the x values are 3 and 5. The difference between them is 2. Since it moves to the right, it’s considered positive.

Step 3: Find the slope. Divide the rise by the run.                                                                   In my example, the rise was 11 and the run was 2 and those divided equals 5.5 which is the slope. Or you can leave it as a fraction.

In Math 10 this week we stated a new unit called Relation and Functions. Relations are when the the independent variable, x, has multiple connections to dependent variables, y. Functions are when each independent variable has a special connection to one dependent variable.

Often written as f(x), if f(x) = 3x + 5

f(9) = 3(9) + 5

f(9) = 32, where the x variable, 9, has the pair of 32 as the y variable                                      on a graph. This would be a point of the graph. Another way of putting                                      this in words is that 9 is the domain and 32 is the range.

Domain is the set of all the numbers for the independent variable in the relation or function.    Range is the set of all numbers for the dependent variable in the relation or function.            So if we were given the domain { -2, -1, 0, 1, 2 } and the function f(x) = 3x^2 -4 then we can find the range { 8, -4, -1} because all we have to do is replace x in the expression with each number of the domain individually and solve.

These have a connection to the last unit we learned with expressions because often there are inputs that are integers and the output are expressions we were introduced to before.

This week in Math 10 we learned how to factor ugly polynomials (polynomials that aren’t able to be factored with other patterns). There is a simple process to follow and finding the factors of ugly polynomials.

Expression: 12x^2 + 17x + 6

Step 1- Go through the list of patterns and choose which one relates to your expression

Common – any common factors within the expression

Difference of Squares – a squared number subtracted from another squared number

Pattern – does it follow the pattern of polynomial that can be factored (x^2 + x + #)

Easy- no leading coefficients

Ugly- don’t follow any of the patterns

Step 2- Take the leading coefficient and the constant of the expression, multiply them together, and find all the possible ways it can be the product of different numbers.

Step 3- Make a box with 4 squares and put the first term in the top left square and the constant in the bottom right square. 

Step 4- Go through the list of the numbers you created and find the one that equals the middle term. If the sign before the constant is a negative then the pairs are different signs. If the the sign before the middle term is a negative, the bigger number of the pair is a negative and if it is a positive then the bigger number is positive. If the sign before the constant is a positive that indicates that both numbers are the same sign.

Step 5- Put the bigger number and its sign in the top right square and the smaller number in the bottom left. With the box completed, go around the box and take out the Greatest Common Factor (The highest number that divides exactly into two or more numbers) with each row and column. The length and width of the box is the factored form of that expression. 

This week in Math 10 we started to learn how to factor polynomials which means to break down polynomials into simpler terms so that when they are multiplied together, they equal the original polynomial.

Step 1: In this example that I have created I started off by finding the greatest common factor (the biggest number that divides into 

the given numbers) of 14, 21, and 35. I did this buy finding the prime factorization of these three numbers and found the common number which is 7.

Step 2: Then I looked for the greatest common factor of the variables and their exponents by looking at which variables I had which were a and b. Then I looked at the smallest exponent of these variables which were 2 for both a and b

GMF of this equation is 7a^2b^2

Step 3: Then I wanted to be able to use this gmf to equal the original polynomial so I had to find out how many times to multiply this gmf to equal all the terms

I ended up with 7a^2b^2(2+3a-5b) 

This week in Math 10 we were introduced to polynomials, but only reviewed what we were taught in grade 9. Some things I remember are models, degrees, types, and distributive property,

Example: 5(8x – 3y) + 2(4y + x)

Step 1: Multiply 5 by 8x and then again with -3y

Ex. 40x – 15y + 2(4y + x)

Step 2: Mutiply 2 by 4y and then again with x

Ex. 40x – 15y + 8y + 2x

Step 3: Collect like terms. Add 2x to 40x and add 8y to -15y

Ex. 42x – 7y

FInal Answer- 42x – 7y