highschool/Grade 9/Math/MPM1DZ/Unit 6: System of Equations.md

# Unit 6: System of Equations 

## Linear System
- Two or more equation that you are working on all together at once on the same set of axes.    
- The lines may ```cross``` or ```intersect``` at a point called the ```Point of Intersection (POI)```.    
- The coordinated of the ```POI``` must satisfy the equation of all the lines in a linear equation.    

- In business, the ```Point of Intersection``` is known as the **Break Even Point** where ```Revenue - Cost = Profit```    
- when **Profit = 0**. There is no gain or loss.

### Number of Solutions
- <img src="https://lh5.googleusercontent.com/wqYggWjMVXvWdY9DiCFYGI7XSL4fXdiHsoZFkiXcDcE93JgZHzPkWSoZ6f4thJ-aLgKd0cvKJutG6_gmmStSpkVPJPOyvMF4-JcfS_hVRTdfuypJ0sD50tNf0n1rukcLBNqOv42A" width="500">   

## 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: $`y = ax^2 + bx + c`$    
 $`
 D = b^2 - 4ac 
 \begin{cases}
 \text{2 distinct real solutions}, & \text{if } D > 0 \\
 \text{1 real solution}, & \text{if } D = 0 \\
 \text{no real solutions}, &  \text{if } D < 0 
 \end{cases}
 `$

- <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">

## Solving Linear-Quadratic Systems
- To find the point of intersection, do the following:     
  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
  5. Use zero-product property to solve for all possible x-values
  6. Subsitute the x-values to one of the original equations to solve for all y-values
  7. State a conclusion / the solution    

- <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">      

- There are 3 possible cases     
- In addition, to determine the number of solutions, you the Discriminant formula $`D = b^2 - 4ac`$


# Ways to solve Systems of Equations
 ## 1. Subsitution   
   - Here we eliminate a variable by subbing in another variable from another equation   
   - We usually do this method if a variable is easily isolated   
   - Example:
     - ```
       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
       ```

 ## 2. Elimination
   - Here we eliminate a variable by basically eliminate a variable from an equation   
   - We usually use this method first when the variables are not easily isolated, then use subsitution to solve   
   - Example:
     - ```
       2x + 3y = 10 (1)
       4x + 3y = 14 (2)
       ```
     - We can then use elimination
       ```
       4x + 3y = 14
       2x + 3y = 10
       ------------
       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
       ```   

## 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

## Solving Systems of Linear Inequalities
- Find the intersection region as the ```solution```.   
- ## If

 - |  |Use ```Dash``` line|Use ```Solid line```|   
   |:-|:------------------|:-------------------|   
   |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   

  - |$`x > a`$ <br> $`x \ge a`$|
    |:------------------| 
    |shade the region on the **right**|  

- ## If   

  - |$`x < a`$ <br> $`x \le a`$|   
    |:------------------|   
    |shade the region on the **left**|  

- Step 1. change all inequalities to ```y-intercept form```   
- Step 2. graph the line   
- Step 3. shade the region where all the regions overlap   

- <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/graphing-systems-of-linear-inequalities/image3.gif" width="400"> 


## Tips
- Read the questions carefully and model the system of equations correctly   
- Be sure to name your equations   
- Label your lines