diff --git a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md
index 14a0c1b..0b78b55 100644
--- a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md
+++ b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md
@@ -136,9 +136,9 @@
## Scientific Notation
- They convey accuracy and precision. It can either be written as its original number or in scientific notation:
-- 555 (**Exact**) or 5.55 x 102 (**3 significant figures**).
-- In scientific notation, values are written in the form **a(10n)**, where ```a``` is a number within 1 and 10 and ```n``` is any integer.
-- Some examples include the following: 5.4 x 103, 3.0 x 102, and 4.56 x 10-4.
+- 555 (**Exact**) or $`5.55 \times 10^2`$ (**3 significant figures**).
+- In scientific notation, values are written in the form $`a(10^n)`$, where $`a`$ is a number within 1 and 10 and $`n`$ is any integer.
+- Some examples include the following: $`5.4 \times 10^3, 3.0 \times 10^2`$, and $`4.56 \times 10^{-4}`$.
- When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used
- 
@@ -340,20 +340,20 @@ x, & \text{if } x > 0\\
## 2D Geometry Equations
|Shape|Formula|Picture|
|:----|:------|:------|
- |Rectangle|```Area```: lw
```Perimeter```: 2(l+w)|
|
- |Triangle|```Area```: bh/2
```Perimeter```: a+b+c|
|
- |Circle|```Area```: πr2
```Circumference```: 2πr or πd|
|
- |Trapezoid|```Area```: (a+b)h/2
```Perimeter```: a+b+c+d|
|
+ |Rectangle|```Area```: $`lw`$
```Perimeter```: $`2(l+w)`$||
+ |Triangle|```Area```: $`\frac{bh}{2}`$
```Perimeter```: $`a+b+c`$||
+ |Circle|```Area```: $`πr^2`$
```Circumference```: $`2πr`$ or $`πd`$||
+ |Trapezoid|```Area```: $` \frac{(a+b)h}{2}`$
```Perimeter```: $`a+b+c+d`$||
## 3D Geometry Equations
|3D Object|Formula|Picture|
|:----|:------|:------|
- |Rectangular Prism|```Volume```: lwh
```SA```: 2(lw+lh+wh)|
|
- |Square Based Pyramid|```Volume```: 1⁄3b2h
```SA```: 2bs+b2|
|
- |Sphere|```Volume```: 4⁄3πr3
```SA```: 4πr2|
|
- |Cone|```Volume```: 1⁄3πr2h
```SA```: πrs+πr2|
|
- |Cylinder|```Volume```: πr2h
```SA```: 2πr2+2πh|
|
- |Triangular Prism|```Volume```: ah+bh+ch+bl
```SA```: 1⁄2blh|
|
+ |Rectangular Prism|```Volume```: $`lwh`$
```SA```: $`2(lw+lh+wh)`$||
+ |Square Based Pyramid|```Volume```: $`\frac{1}{3} b^2 h`$
```SA```: $`2bs+b^2`$||
+ |Sphere|```Volume```: $`\frac{4}{3} πr^3`$
```SA```: $`4πr^2`$||
+ |Cone|```Volume```: $` \frac{1}{3} πr^2 h`$
```SA```: $`πrs+πr^2`$||
+ |Cylinder|```Volume```: $`πr^2h`$
```SA```: $`2πr^2+2πh`$||
+ |Triangular Prism|```Volume```: $`ah+bh+ch+bl`$
```SA```: $` \frac{1}{2} blh`$||
## Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)
@@ -362,18 +362,18 @@ x, & \text{if } x > 0\\
|Shape|Maximum Area|Minimum Perimeter|
|:----|:-----------|:----------------|
- |4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width
A = lw
Amax = (w)(w)
Amax = w2|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.
P = 2(l+w)
Pmin = 2(w)(w)
Pmin = 2(2w)
Pmin = 4w|
- |3-sided rectangle|l = 2w
A = lw
Amax = 2w(w)
Amax = 2w2|l = 2w
P = l+w2
Pmin = 2w+2w
Pmin = 4w|
+ |4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width
$`A = lw`$
$`A_{max} = (w)(w)`$
$`A_{max} = w^2`$|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.
$`P = 2(l+w)`$
$`P_{min} = 2(w)(w)`$
$`P_{min} = 2(2w)`$
$`P_{min} = 4w`$|
+ |3-sided rectangle|$`l = 2w`$
$`A = lw`$
$`A_{max} = 2w(w)`$
$`A_{max} = 2w^2`$|$`l = 2w`$
$`P = l+2w`$
$`P_{min} = 2w+2w`$
$`P_{min} = 4w`$|
### 3D Objects
|3D Object|Maximum Volumne|Minimum Surface Area|
|:--------|:--------------|:-------------------|
- |Cylinder(closed-top)|The cylinder must be similar to a cube where h = 2r
V = πr2h
Vmax = πr2(2r)
Vmax = 2πr3|The cylinder must be similar to a cube where h = 2r
SA = 2πr2+2πrh
SAmin = 2πr2+2πr(2r)
SAmin = 2πr2+4πr2
SAmin = 6πr2|
- |Rectangular Prism(closed-top)|The prism must be a cube,
where l = w = h
V = lwh
Vmax = (w)(w)(w)
Vmax = w3|The prism must be a cube,
where l = w = h
SA = 2lh+2lw+2wh
SAmin = 2w2+2w2+2w2
SAmin = 6w2|
- |Cylinder(open-top)|h = r
V = πr2h
Vmax = πr2(r)
Vmax = πr3|h = r
SA = πr2+2πrh
SAmin = πr2+2πr(r)
SAmin = πr2+2πr2
SAmin = 3πr2|
- |Square-Based Rectangular Prism(open-top)|h = w/2
V = lwh
Vmax = (w)(w)(w⁄2)
Vmax = w3⁄2|h = w/2
SA = w2+4wh
SAmin = w2+4w(w⁄2)
SAmin = w2+2w2
SAmin = 3w2|
+ |Cylinder(closed-top)|The cylinder must be similar to a cube where $`h = 2r`$
$`V = πr^2h`$
$`V_{max} = πr^2(2r)`$
$`V_{max} = 2πr^3`$|The cylinder must be similar to a cube where $`h = 2r`$
$`SA = 2πr^2+2πrh`$
$`SA_{min} = 2πr^2+2πr(2r)`$
$`SA_{min} = 2πr^2+4πr^2`$
$`SA_{min} = 6πr^2`$|
+ |Rectangular Prism(closed-top)|The prism must be a cube,
where $`l = w = h`$
$`V = lwh`$
$`V_{max} = (w)(w)(w)`$
$`V_{max} = w^3`$|The prism must be a cube,
where $`l = w = h`$
$`SA = 2lh+2lw+2wh`$
$`SA_{min} = 2w^2+2w^2+2w^2`$
$`SA_{min} = 6w^2`$|
+ |Cylinder(open-top)|$`h = r`$
$`V = πr^2h`$
$`V_{max} = πr^2(r)`$
$`V_{max} = πr^3`$|$`h = r`$
$`SA = πr^2+2πrh`$
$`SA_{min} = πr^2+2πr(r)`$
$`SA_{min} = πr^2+2πr^2`$
$`SA_{min} = 3πr^2`$|
+ |Square-Based Rectangular Prism(open-top)|$`h = \frac{w}{2}`$
$`V = lwh`$
$`V_{max} = (w)(w)(\frac{w}{2})`$
$`V_{max} = \frac{w^3}{2}`$|$`h = \frac{w}{2}`$
$`SA = w^2+4wh`$
$`SA_{min} = w^2+4w(\frac{w}{2})`$
$`SA_{min} = w^2+2w^2`$
$`SA_{min} = 3w^2`$|
## Labelling
- Given any polygons, labelling the vertices must always:
@@ -420,10 +420,10 @@ x, & \text{if } x > 0\\
## Slope and Equation of Line
- ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```the rate of change```
-- ```Slope Formula```: **m = (y2-y1)/(x2-x1)**
-- ```Standard Form```: **ax + by + c = 0**, a∈Z, b∈Z, c∈Z (must be integers and ```a``` must be positive)
-- ```Y-intercept Form```: **y = mx + b**
-- ```Point-slope Form```: **y2-y1 = m(x2-x1)**
+- ```Slope Formula```: $`m = \frac{y_2 - y_1}{x_2 - x_1}`$
+- ```Standard Form```: $`ax + by + c = 0, a \isin \mathbb{Z}, b \isin \mathbb{Z}, c \isin \mathbb{Z}`$ (must be integers and $`a`$ must be positive)
+- ```Y-intercept Form```: $`y = mx + b`$
+- ```Point-slope Form```: $`y_2-y_1 = m(x_2-x_1)`$
- The slope of a vertical lines is undefined
- The slope of a horizontal line is 0
- Parallel lines have the ```same slope```
@@ -432,25 +432,25 @@ x, & \text{if } x > 0\\
## Relations
- A relation can be described using
1. Table of Values (see below)
- 2. Equations (y = 3x + 5)
+ 2. Equations $`(y = 3x + 5)`$
3. Graphs (Graphing the equation)
4. Words
- When digging into the earth, the temperature rises according to the
-- following linear equation: t = 15 + 0.01 h. **t** is the increase in temperature in
-- degrees and **h** is the depth in meters.
+- following linear equation: $`t = 15 + 0.01 h`$. $`t`$ is the increase in temperature in
+- degrees and $`h`$ is the depth in meters.
## Perpendicular Lines
- To find the perpendicular slope, you will need to find the slope point
- Formula: slope1 × slope2 = -1
-- Notation: m⊥
+- Notation: $`m_\perp`$
- 
## Definitions
- ```Parallel```: 2 lines with the same slope
- ```Perpendicular```: 2 lines with slopes that are the negative reciprocal to the other. They form a 90 degree angle where they meet.
-- ```Domain```: The **ordered** set of all possible values of the independent variable (x).
-- ```Range```: The **ordered** set of all possible values of the dependent variable (y).
+- ```Domain```: The **ordered** set of all possible values of the independent variable $`x`$.
+- ```Range```: The **ordered** set of all possible values of the dependent variable $`y`$.
- ```Continous Data```: A data set that can be broken into smaller parts. This is represented by a ```Solid line```.
- ```Discrete Data```: A data set that **cannot** be broken into smaller parts. This is represented by a ```Dashed line```.
- ```First Difference```: the difference between 2 consecutive y values in a table of values which the difference between the x-values are constant.
@@ -525,18 +525,22 @@ x, & \text{if } x > 0\\
- 
## Discriminant
-- The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the
-- coefficients and constant of a quadratic equation: **y = ax2 + bx + c**
-- D = b2 - 4ac
-- D > 0 ```(2 distinct real solutions)```
-- D = 0 ```(1 real solution)```
-- D < 0 ```(no real solutions)```
+- The discriminant determines the number of solutions (roots) there are in a quadratic equation. $`a, b , c`$ are the
+- coefficients and constant of a quadratic equation: $`y = ax^2 + bx + c`$
+ $`
+ D = b^2 - 4ac
+ \begin{cases}
+ \text{2 distinct real solutions}, & \text{if } D > 0 \\
+ \text{1 real solution}, & \text{if } D = 0 \\
+ \text{no real solutions}, & \text{if } D < 0
+ \end{cases}
+ `$
- 
## Solving Linear-Quadratic Systems
- To find the point of intersection, do the following:
- 1. Isolate both equations for ```y```
+ 1. Isolate both equations for $`y`$
2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
3. Simplify and put everything on one side and equal to zero on the other side
4. Factor
@@ -547,7 +551,7 @@ x, & \text{if } x > 0\\
- 
- There are 3 possible cases
-- In addition, to determine the number of solutions, you the Discriminant formula **D = b2 - 4ac**
+- In addition, to determine the number of solutions, you the Discriminant formula $`D = b^2 - 4ac`$
# Ways to solve Systems of Equations
@@ -559,14 +563,18 @@ x, & \text{if } x > 0\\
y = x + 10 (1)
x + y + 34 = 40 (2)
```
- We can sub (1) into (2) to find ```x```, then you the value of ```x``` we found to solve for ```y```
- ```x + (x + 10) + 34 = 40```
- ```2x + 44 = 40```
- ```2x = -4```
- ```x = -2```
- Then solve for ```y```
- ```y = -2 + 10```
- ```y = -8```
+ - We can sub $`(1)`$ into $`(2)`$ to find $`x`$, then you the value of $`x`$ we found to solve for $`y`$
+ ```
+ x + (x + 10) + 34 = 40
+ 2x + 44 = 40
+ 2x = -4
+ x = -2
+ ```
+ - Then solve for $`y`$
+ ```
+ y = -2 + 10
+ y = -8
+ ```
## 2. Elimination
- Here we eliminate a variable by basically eliminate a variable from an equation
@@ -576,7 +584,7 @@ x, & \text{if } x > 0\\
2x + 3y = 10 (1)
4x + 3y = 14 (2)
```
- We can then use elimination
+ - We can then use elimination
```
4x + 3y = 14
2x + 3y = 10
@@ -584,10 +592,12 @@ x, & \text{if } x > 0\\
2x + 0 = 4
x = 2
```
- Then sub the value of ```x``` into an original equation and solve for ```y```
- ```2(2) + 3y = 10```
- ```3y = 6```
- ```y = 2```
+ - Then sub the value of $`x`$ into an original equation and solve for $`y`$
+ ```
+ 2(2) + 3y = 10
+ 3y = 6
+ y = 2
+ ```
## 3. Graphing
- we can rewrite the equations into ```y-intercept form``` and then graph the lines, and see where the lines intersect (P.O.I), and the P.O.I is the solution
@@ -598,20 +608,20 @@ x, & \text{if } x > 0\\
- | |Use ```Dash``` line|Use ```Solid line```|
|:-|:------------------|:-------------------|
- |Shade the region ```above``` the line|y > mx + b|y ≥ mx + b|
- |Shade the region ```below``` the line|y < mx + b| y ≤ mx + b|
+ |Shade the region ```above``` the line|$`y > mx + b`$|$`y \ge mx + b`$|
+ |Shade the region ```below``` the line|$`y < mx + b`$|$`y \le mx + b`$|
- ## If
- - |x > a
x ≥ a|
+ - |$`x > a`$
$`x \ge a`$|
|:------------------|
- - |shade the region on the **right**|
+ |shade the region on the **right**|
- ## If
- - |x < a
x ≤ a|
- - |:------------------|
- - |shade the region on the **left**|
+ - |$`x < a`$
$`x \le a`$|
+ |:------------------|
+ |shade the region on the **left**|
- Step 1. change all inequalities to ```y-intercept form```
- Step 2. graph the line