Throughout this week in Math 9 we have been working on linear equations and the rules that go along with them. For this assignment, I will explain how to do linear equations with variables and constants on both sides.

To do linear equations, you need to remember a few key steps and rules:
1. You want to move the variables to 1 side of the equation, and the constants to the other. You can do this for which ever side you find easier (ex moving the smaller variable so you don’t have to deal with a negative variable)

2. When moving anything across the = sign, turn it into the inverse of itself (negatives to positive, positive to negative).

3. Add everything on each individual side together.

4. Divide the constant (on the other side of the equals sign) by the coefficient of the variable (if necessary). This will result in the answer for 1 of the variable (ex. 1y). However, if the coefficient was negative, it will result in the answer for -1y, in which case you need to turn it into the inverse of itself.

5. Double check your answer by going to the original question, and plugging the answer in, replacing the variable. See if it checks out.

Example 1:
2x = x + 1
2x – x = 1 (Moving x across the equals sign makes it negative)
x = 1 (Add everything on each individual side together)

Verifying example 1:
2x = x + 1
2*1 = 1 + 1
2 = 1 + 1
2 = 2

Example 2:
6x + 10 – 4x = 12x – 20
10 + 20 = 10x (Moving -20 across the equals sign made it positive, moving 6x made it negative, moving -4x made it positive. All the constants are now on one side, and all the variables are on the other)
30 = 10x (Add everything on each individual side together)
30/10 = 3 (Divide the constant by the coefficient, the coefficient is positive so we now have the result for positive 1x)
x = 3

Verifying example 2:
6x + 10 – 4x = 12x – 20
6*3 + 10 – 4*3 = 12*3 – 20
18 + 10 – 12 = 36 – 20
16 = 16

Throughout this week in Math 9 we have been working on powers and the rules that go along with them. For this assignment, I will explain the product rule and the exponent rule.

The product rule states that if two numbers are being multiplied that have the same base, you can simplify it by adding together the powers. Remember that if a number has no power, it actually has an invisible 1 power.

Example 1: 4^4 * 4^6 = 4^10
Example 2: 3 * 3^3 = 3^1 * 3^3 = 3^4

The exponent rule states that to simplify an expression where there are powers inside and outside a bracket, you can multiply the exponents (known as raising the power to the other power).

Example 1: (4^4)^4 = 4^16
Example 2: (8^3)^7 = 8^21

Throughout this week in Math 9 we have been working on adding and substracting fractions. For this assignment I will explain how to add fractions.

There are 4 main steps to adding fractions:
1. Turn any mixed fractions into improper fractions by multiplying the whole number by the denominator, and adding the sum to the nominator.
2. Find the LCM (lowest common multiple) of the denominators of the fractions you are multiplying. This is done by multiplying the denominator on each number by 1 to start, and raising the number by 1 until there are matching products from the current denominator you multiplied and any of the previous denominators you multiplied from the other number. Keep track of how many multiplications it took to find the LCM of EACH number.

Example: 4 and 6
4: 4(1), 8(2), 12(3)
6: 6(1), 12(2)

LCM:12

3. Change the denominators of both fractions to the LCM and multiply the nominators of both fractions to the number of multiplications it took to reach the LCM.
4. Add the two fraction’s nominators together. If desired/needed, revert to a mixed fraction and simplify the fraction by finding the lowest common divider (opposite of LCM).