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Grade 9/Math.md
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Grade 9/Math.md
@ -53,7 +53,7 @@
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>> | a ∩ b | Intersection (and) |
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> ## Pythgorean Theorem
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>> let a be the adjecant and b be the opposite, and c be the hypotenuse.
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>> a and b are the two legs of the triangle or two sides that form a 90 degree angle of the triangle, c is the hypotenuse
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>> a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>
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>> <img src="https://docs.google.com/drawings/u/1/d/sGjyHDIs-wHWzppHAGdIpEA/image?w=162&h=70&rev=1&ac=1&parent=1ZIXKcDk3LBlgPK2EoUV04c0G1LZotrtfgVhJTooO1zA" width="200">
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@ -68,10 +68,10 @@
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>> To Multiply rationals, first reduce the fraction to their lowest terms, then multiply numerators and denominators
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>> To Divide rationals, multiply them by the reciprocal
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>> ### Example Simplify Fully:
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>>> = <sup>3</sup>⁄<sub>4</sub> ÷ <sup>2</sup>⁄<sub>12</sub> [Reduce to lowest terms]
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>>> = <sup>3</sup>⁄<sub>4</sub> ÷ <sup>1</sup>⁄<sub>7</sub> [Multiply by reciprocal]
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>>> = <sup>3</sup>⁄<sub>4</sub> × 7
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>>> = <sup>21</sup>⁄<sub>4</sub> [Leave as an improper fraction]
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>>> = <a href="https://www.codecogs.com/eqnedit.php?latex==&space;\frac{3}{4}&space;\div&space;\frac{2}{14}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?=&space;\frac{3}{4}&space;\div&space;\frac{2}{14}" title="= \frac{3}{4} \div \frac{2}{14}" /></a>[Reduce to lowest terms]
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>>> = <a href="https://www.codecogs.com/eqnedit.php?latex=\frac{3}{4}&space;\div&space;\frac{1}{7}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{3}{4}&space;\div&space;\frac{1}{7}" title="\frac{3}{4} \div \frac{1}{7}" /></a> [Multiply by reciprocal]
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>>> = <a href="https://www.codecogs.com/eqnedit.php?latex=\frac{3}{4}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{3}{4}" title="\frac{3}{4}" /></a> × 7
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>>> = <a href="https://www.codecogs.com/eqnedit.php?latex=\frac{21}{4}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{21}{4}" title="\frac{21}{4}" /></a> [Leave as an improper fraction]
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>> ### Shortcut for multiplying fractions
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>>> cross divide to keep your numbers small
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@ -93,7 +93,7 @@
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>>> |Rational Exponents|a<sup>n/m</sup> = (<sup>m</sup>√a)<sup>n</sup>|16<sup>5/4</sup> = (<sup>4</sup>√16)<sup>5</sup> = 2<sup>5</sup>|
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>>> **Note:**
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>>> Standard --> Expanded Form
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>>> Exponential Form --> Expanded Form
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>>> 6<sup>4</sup> = 6 × 6 × 6 × 6
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>> ## Scientific Notation
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@ -188,26 +188,16 @@
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>>> 1. Expand and simplify both sides
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>>> 2. Isolate using reverse order of operations
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>>> 3. Check the solution by plugging the variable back into the equation and check if the ```left side``` equals the ```right side```
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> ## Venn Diagrams
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>> <img src="https://d2slcw3kip6qmk.cloudfront.net/marketing/blog/2017Q3/Venn-diagram-symbols-and-notation/VDIntersections.png" width="400">
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>> ```Set```: a collection of elements, O (the circle)
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>> ```Universal Set```: This is a collection of all the elements that you are interested in. Use ```{}``` bracket to write the set inside the rectangle
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>> ```Union```: ∪
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>> if 2 sets have union, all the elements belong to any of the set. This is known as ```or```
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>> ```Intersection```: ∩
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>> if 2 sets have an intersection, they are elements belonging to both sets. This is known as ```and```
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>> ```Set Notation```:
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>> A notation that represents the collection of numbers. It is written in this form x = {x|x∈R}
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>> ```Absolute Value```
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>>> There are 2 cases. For this sort of equation, you must split the equation into 2 separate equations. One of the
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>>> equations will have the absolute bracket be positive while the other negative.
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>>> Absolute values are written in the form ```| x |```
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>>> where
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>>> if x > 0, | x | = x
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>>> if x == 0, | x | = 0
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>>> if x < 0, | x | = -x
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>
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> ## Absolute Values
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>> There are 2 cases. For this sort of equation, you must split the equation into 2 separate equations. One of the
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>> equations will have the absolute bracket be positive while the other negative.
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>> Absolute values are written in the form ```| x |```
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>> where
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>> if x > 0, | x | = x
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>> if x = 0, | x | = 0
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>> if x < 0, | x | = -x
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> ## Quadractic Equations
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>> ```Quadratic Function```: A parabolic graph where the axis of symmetry is parallel to the y-axis
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@ -225,31 +215,7 @@
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>> Note:
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>> √x<sup>2</sup> = ± x (There are 2 possible solutions)
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>> ```Distrubutive Property``` - This is opening the bracket. a(x+y) = ax+ay
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>> ## Discriminant
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>>> The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the
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>>> coefficients and constant of a quadratic equation: ```y = ax<sup>2</sup> + bx + c```
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>>> D = b<sup>2</sup> - 4ac
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>>> D > 0 ```(2 distinct real solutions)```
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>>> D = 0 ```(1 real solution)```
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>>> D < 0 ```(no real solutions)```
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>>> <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">
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> ## Solving Linear-Quadratic Systems
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>> To find the point of intersection, do the following:
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>> 1. Isolate both equations for ```y```
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>> 2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
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>> 3. Simplify and put everything on one side and equal to zero on the other side
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>> 4. Factor
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>> 5. Use zero-product property to solve for all possible x-values
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>> 6. Subsitute the x-values to one of the original equations to solve for all y-values
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>> 7. State a conclusion / the solution
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> <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">
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> There are 3 possible cases
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> In addition, to determine the number of solutions, you the Discriminant formula **D = b<sup>2</sup> - 4ac**
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> ## Tips
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>> ```Absolute Values``` can have 2 solutions
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>> ```Quadratics``` can also have 2 solutions
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@ -292,10 +258,22 @@
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> <img src="https://www.katesmathlessons.com/uploads/1/6/1/0/1610286/exterior-angle-theorem-diagram-picture_orig.png" width="300">
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> 7. ``` Isosceles Triangle Theorem``` (ITT)
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> - The base angles in any isosceles triangle are equal
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> - The base angles in any isosceles triangle are equal
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> <img src="http://www.assignmentpoint.com/wp-content/uploads/2016/06/isosceles-triangle-theorem.jpg" width="400">
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> 8. ```Sum of The Interior Angle of a Polygon```
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> - The sum of the interioir angles of any polygon is ```180(n-2)``` or ```180n - 360```, where ```n``` is the number of sides of the polygon
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<img src="https://i.ytimg.com/vi/tmRpwCM1K1o/maxresdefault.jpg" width="500">
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> 9. ```Exterior Angles of a Convex Polygon```
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> - The sum of the exterior angle of any convex polygon is always ```360 degrees```
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<img src="https://image.slidesharecdn.com/findanglemeasuresinapolygon-110307143453-phpapp02/95/find-angle-measures-in-a-polygon-11-728.jpg?cb=1299508555" width="400">
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> ## Properties of Quadrilaterals
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>> Determine the shape using the properties of it
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> ```Linear Relation```: A relation which a single straight line can be drawn through every data point and the first differences are constant
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> ```Non - Linear Relation```: A single smooth curve can be drawn through every data point and the first differences are not constant
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> ## Slope and Equation of Line
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>> ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```change in y / change in x```
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>> ```Slope Formula```: **m = y<sub>2</sub>-y<sub>1</sub>/x<sub>2</sub>-x<sub>1</sub>**
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>> ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```rate of change y / rate of change x```
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>> ```Slope Formula```: **m = (y<sub>2</sub>-y<sub>1</sub>)/(x<sub>2</sub>-x<sub>1</sub>)**
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>> ```Standard Form```: **ax + by + c = 0**, a∈Z, b∈Z, c∈Z (must be integers and ```a``` must be positive)
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>> ```Y-intercept Form```: **y = mx + b**
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>> ```Point-slope Form```: **y<sub>2</sub>-y<sub>1</sub> = m(x<sub>2</sub>-x<sub>1</sub>)**
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>> following linear equation: t = 15 + 0.01 h. **t** is the increase in temperature in
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>> degrees and **h** is the depth in meters.
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> ## Perpendicular Bisector
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>> To find the perpendicular bisector, you will need to fidn the slope and midpoint
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> ## Perpendicular Lines
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>> To find the perpendicular slope, you will need to find the slope point
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>> Formula: slope1 × slope2 = -1
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>> Notation: m<sub>⊥</sub>
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>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/parallel-perpendicular-lines/parallel_perpendicular_lines_1.gif" width="300">
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@ -446,13 +424,24 @@
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>> You can't go backwards on the x-axis, as you can't go back in time
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>> Plot the points accordingly
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>> Draw the lines accordingly
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>> <img src="https://dryuc24b85zbr.cloudfront.net/tes/resources/6061038/image?width=500&height=500&version=1519313844425" width="400">
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>> <img src="https://dryuc24b85zbr.cloudfront.net/tes/resources/6061038/image?width=500&height=500&version=1519313844425" width="400">
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>> **Direction is always referring to:**
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>> 1. ```go towards home```
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>> 2. ```going away from home```
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>> 3. ```stop```
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> ## Scatterplot and Line of Best Fit
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>> A scatterplot graph is there to show the relation between two variables in a table of values.
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>> A line of best fit is a straight line that describes the relation between two variables.
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>> If you are drawing a line of best fit, try to use as many data points, have an equal amount of points onto and under the line of best fit, and keep it as a straight line.
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>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/line%20of%20best%20fit-eyeball/lineofbestfit-e-1.gif" width="300">
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>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/line%20of%20best%20fit-eyeball/lineofbestfit-e-1.gif" width="300">
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>> ### How To Determine the Equation Of a Line of Best Fit
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>> 1. Find two points **```ON```** the ```line of best fit```
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>> 2. Determine the ```slope``` using the two points
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>> 3. Use ```point-slope form``` to find the equation of the ```line of best fit```
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> ## Table of values
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>> To find first differences or any points on the line, you can use a ```table of values```
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@ -486,6 +475,32 @@
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>> ### Number of Solutions
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>>> <img src="https://lh5.googleusercontent.com/wqYggWjMVXvWdY9DiCFYGI7XSL4fXdiHsoZFkiXcDcE93JgZHzPkWSoZ6f4thJ-aLgKd0cvKJutG6_gmmStSpkVPJPOyvMF4-JcfS_hVRTdfuypJ0sD50tNf0n1rukcLBNqOv42A" width="500">
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> ## Discriminant
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>> The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the
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>> coefficients and constant of a quadratic equation: ```y = ax<sup>2</sup> + bx + c```
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>> D = b<sup>2</sup> - 4ac
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>> D > 0 ```(2 distinct real solutions)```
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>> D = 0 ```(1 real solution)```
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>> D < 0 ```(no real solutions)```
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>> <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">
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> ## Solving Linear-Quadratic Systems
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>> To find the point of intersection, do the following:
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>> 1. Isolate both equations for ```y```
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>> 2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
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>> 3. Simplify and put everything on one side and equal to zero on the other side
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>> 4. Factor
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>> 5. Use zero-product property to solve for all possible x-values
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>> 6. Subsitute the x-values to one of the original equations to solve for all y-values
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>> 7. State a conclusion / the solution
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>> <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">
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>> There are 3 possible cases
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>> In addition, to determine the number of solutions, you the Discriminant formula **D = b<sup>2</sup> - 4ac**
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> # Ways to solve Systems of Equations
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> 1. Subsitution
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> Here we eliminate a variable by subbing in another variable from another equation
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