From 31b126657d9fd8463a05712115e26d7e72545d53 Mon Sep 17 00:00:00 2001 From: James Su Date: Wed, 4 Sep 2019 23:39:22 +0000 Subject: [PATCH] Add new file --- .../MPM1DZ/Unit 6: System of Equations.md | 146 ++++++++++++++++++ 1 file changed, 146 insertions(+) create mode 100644 Grade 9/Math/MPM1DZ/Unit 6: System of Equations.md diff --git a/Grade 9/Math/MPM1DZ/Unit 6: System of Equations.md b/Grade 9/Math/MPM1DZ/Unit 6: System of Equations.md new file mode 100644 index 0000000..6e077d8 --- /dev/null +++ b/Grade 9/Math/MPM1DZ/Unit 6: System of Equations.md @@ -0,0 +1,146 @@ + +# 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 +- + +## 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: $`y = ax^2 + bx + c`$ + $` + D = b^2 - 4ac + \begin{cases} + \text{2 distinct real solutions}, & \text{if } D > 0 \\ + \text{1 real solution}, & \text{if } D = 0 \\ + \text{no real solutions}, & \text{if } D < 0 + \end{cases} + `$ + +- + +## 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 + +- + +- 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 + + - |$`x > a`$
$`x \ge a`$| + |:------------------| + |shade the region on the **right**| + +- ## If + + - |$`x < a`$
$`x \le a`$| + |:------------------| + |shade the region on the **left**| + +- Step 1. change all inequalities to ```y-intercept form``` +- Step 2. graph the line +- Step 3. shade the region where all the regions overlap + +- + + +## Tips +- Read the questions carefully and model the system of equations correctly +- Be sure to name your equations +- Label your lines + +# General Tips +- Be sure to watch out for units, like ```cm``` or ```km``` +- Watch out for ```+/-``` +- Be sure to reverse the operation when moving things to the other side of the equation +- Make sure to have a proper scale for graphs +- Read question carefully and use the appropriate tools to solve +- **WATCH OUT FOR CARELESS MISTAKES!!!!!!!!!!!** + +## Word Problems +- Read carefully +- model equations correctly +- ```Reread``` the question over and over again until you fully understand it and made sure there is no tricks. :p +- ```Lets``` Statement +- ```Conclusion``` + +## Graph Problems +- Look up on tips in units (5) and (6) +- be sure to use a ruler when graphing + +## System of Equations +- When in doubt or to check your work, just plug the numbers back in and check if the statement is true \ No newline at end of file