diff --git a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md deleted file mode 100644 index 623a2d9..0000000 --- a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md +++ /dev/null @@ -1,663 +0,0 @@ -# Math Study Sheet!!!! - -# Exam Detail -|Unit|Marks| -|:---|:----| -|Unit 1|10| -|Unit 2|10| -|Unit 3|9| -|Unit 4|11| -|Unit 5|11| -|Unit 6|8| -|Forms|4| -|Total|63| - -|Section|Marks| - |:------|:----| - |Knowledge|21| - |Application|23| - |Thinking|12| - |Communication|3| - |Forms|4| - - |Part|Question| - |:---|:-------| - |A|9 multiple choice| - |B|10 Short Answer -->
- 7 Knowledge questions
- 3 Application Questions| - |C|10 Open Response -->
- 10 Knowledge Questions
- 5 Application Questions
- 3 Thinking Questions
- 1 Communication Question| - -# Unit 1: Essential Skills - -## Simple Arithmetics - -### Addition / Subtraction - | Expression | Equivalent| - |:----------:|:---------:| - | a + b | a + b | - | (-a) + b | b - a | - | a + (-b) | a - b | - | (-a) + (-b) | -(a + b) | - | a - b | a - b| - | a - (-b) | a + b | - | (-a) -(-b) | (-a) + b| - -### Multiplication / Division - | Signs | Outcome | - |:-----:|:-------:| - | a * b | Positive | - | (-a) * b | Negative | - | a * (-b) | Negative | - | (-a) * (-b) | Positive | - -### BEDMAS / PEMDAS -- Follow ```BEDMAS``` for order of operations if there are more than one operation - - | Letter | Meaning | - |:------:|:-------:| - | B / P | Bracket / Parentheses | - | E | Exponent | - | D | Divison | - | M | Multiplication | - | A | Addition | - | S | Subtraction | - -- - -## Interval Notation -- A notation that represents an interval as a pair of numbers. -- The numbers in the interval represent the endpoint. E.g. **[x > 3, x ∈ R]** -- ```|``` means ```such that``` -- ```E``` or ∈ means ```element of``` -- ```N``` represents **Natural Numbers** $`N = \{x | x \gt 0, x \isin \mathbb{Z} \}`$ -- ```W``` represents **Whole Numbers** $`W = \{x | x \ge 0, x \isin \mathbb{Z}\}`$ -- ```Z``` represents **Integers** $`Z = \{x| -\infin \le x \le \infin, x \isin \mathbb{Z}\}`$ -- ```Q``` represents **Rational Numbers** $`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`$ - - | Symbol | Meaning | - |:------:|:-------:| - | (a, b) | Between but not including ```a``` or ```b```, you also use this for ```∞``` | - | [a, b] | Inclusive | - | a ∪ b | Union (or) | - | a ∩ b | Intersection (and) | - -## Pythgorean Theorem -- 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 -- $`a^2+b^2=c^2`$ - -- - -## Operations with Rationals -- $`Q = \{ \frac{a}{b} |a, b \isin \mathbb{Z}, b \neq 0 \}`$ - -- Any operations with rationals, there are 2 sets of rules - 1. ```Rules for operations with integers``` - 2. ```Rules for operations with fractions``` - -- To Add / subtract rationals, find common denominator and then add / subtract numerator -- To Multiply rationals, first reduce the fraction to their lowest terms, then multiply numerators and denominators -- To Divide rationals, multiply them by the reciprocal - -### Example Simplify Fully: - -- $` \frac{3}{4} \div \frac{2}{14} `$ Reduce to lowest terms - -- $` \frac{3}{4} \div \frac{1}{7} `$ Multiple by reciprocal - -- $` \frac{3}{4} \times 7 `$ - -- $` = \frac{21}{4}`$ Leave as improper fraction - - -### Shortcut for multiplying fractions -- cross divide to keep your numbers small -- Example: -- $` \frac{3}{4} \times \frac{2}{12} `$ - -- $` \frac{1}{2} \times \frac{1}{4} `$ - -- $` = \frac{1}{8} `$ - - -## Exponent Laws - - | Rule | Description| Example | - |:----:|:----------:|:-------:| - |Product|am × an = an+m|23 × 22 = 25| - |Quotient|am ÷ an = an-m|34 ÷ 32 = 32| - |Power of a Power|(am)n = amn|(23)2 = 26| - |Power of a Quotient| = | = | - |Zero as Exponents|a0 = 1|210 = 1| - |Negative Exponents|a-m = |1-10 = | - |Rational Exponents|an/m = | = | - -**Note:** -- Exponential Form --> Expanded Form -- 64 = 6 × 6 × 6 × 6 - -## Scientific Notation -- They convey accuracy and precision. It can either be written as its original number or in scientific notation: -- 555 (**Exact**) or $`5.55 \times 10^2`$ (**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 - -- - -- **Remember**: For scientific notation, round to ```3 significant``` digits - -## Rates, Ratio and Percent -- ```Ratio```: A comparison of quantities with the same unit. These are to be reduced to lowest terms. -- Examples: ```a:b, a:b:c, a/b, a to b ``` - -- ```Rates```: A comparison of quantities expressed in different units. -- Example: ```10km/hour``` - -- ```Percent```: A fraction or ratio in which the denominator is 100 -- Examples: ```50%, 240/100``` - -## Number Lines -- a line that goes from a point to another point, a way to visualize set notations and the like -- -- A solid filled dot is used for ```[]``` and a empty dot is used for ```()``` - - -## Tips -- Watch out for the ```+/-``` signs -- Make sure to review your knowledge of the exponent laws -- For scientific notation, watch out for the decimal point -- Use shortcut when multiplying fractions - - -# Unit 2: Polyomials -## Introduction to Polynomials -- A ```variable``` is a letter that represents one or more numbers -- An ```algebraic expression``` is a combination of variables and constants ```(e.g. x+y+6. y + 8)``` -- When a specific value is assigned to a variable in a algebraic expression, this is known as substitution. -## Methods to solve a polynomial - 1. ```Combine like terms``` - 2. ```Dividing polynomials``` - 3. ```Multiplying polynomials``` - -## Simplifying Alegebraic Expressions -- An algebraic expression is an expression with numbers, variables, and operations. You may expand or simplify equations thereon. - -## Factoring -- Two methods of solving; decomposition and criss-cross. First of all, the polynomial must be in the form of a quadratic -- equation (ax2 + bx + c). As well, simplify the polynomial, so that all common factors are outside -- (e.g 5x + 10 = 5(x + 2) ). - -|Type of Polynomial|Definition| - |:-----------------|:---------| - |Monomial|Polynomial that only has one term| - |Binomial|Polynomial that only has 2 terms| - |Trinomial|polynomial that only has 3 terms| - - |Type|Example| - |:--:|:-----:| - |Perfect Square Trinomials| $`(a+b)^2 = a^2+2ab+b^2 or (a-b)^2 = a^2-2ab+b^2`$| - |Difference with Squares|$`a^2-b^2 = (a+b)(a-b)`$| - |Simple Trinomials|$`x^2+6x-7 = (x+7)(x-1)`$| - |Complex Trinomials|$`2x^2-21x-11 = (2x+1)(x-11)`$| - |Common Factor|$`2ab+6b+4 = 2(ab+3b+2)`$| - |Factor By Grouping|$`ax+ay+bx+by = (ax+ay)+(bx+by) = a(x+y)+b(x+y) = (a+b)(x+y)`$| - -## Shortcuts - -- - -## Foil / Rainbow Method -- - -## Definitions -- ```Term``` a variable that may have coefficient(s) or a constant -- ```Alebraic Expressions```: made up of one or more terms -- ```Like-terms```: same variables raised to the same exponent - -## Tips -- Be sure to factor fully -- Learn the ```criss-cross``` (not mandatory but its a really good method to factor quadratics) -- Learn ```long division``` (not mandatory but its a really good method to find factors of an expression) -- Remember your formulas -- Simplify first, combine like terms - -# Unit 3: Solving Equations and Inequailties - -## Equations -- a ```mathematical statement``` in which the value on the ```left side``` equals the value on the ```right side``` of the equal sign -- To ```solve``` and equation is to find the variable that makes the statement true -### Methods to solve an equation - 1. Expand and simplify both sides - 2. Isolate using reverse order of operations - 3. Check the solution by plugging the variable back into the equation and check if the ```left side``` equals the ```right side``` - -## Absolute Values -- There are 2 cases. For this sort of equation, you must split the equation into 2 separate equations. One of the -- equations will have the absolute bracket be positive while the other negative. -- Absolute values are written in the form ```| x |``` -- where -$` -| x | = -\begin{cases} -x, & \text{if } x > 0\\ -0, & \text{if } x = 0\\ --x, & \text{if } x < 0 -\end{cases} -`$ - -## Quadractic Equations -- ```Quadratic Function```: A parabolic graph where the axis of symmetry is parallel to the y-axis -- ```Quadratic Equation```: This function is set equal to ```0```. The solution to the equation are called ```roots``` -- Solve quadratic equation by: - - 1. Isolation - - $`a(x+b)^2 + k = 0`$ - 2. Factor using zero-product property - - ```The Zero Factor Property``` refers to when a×b=0, then either a=0 or b=0. - - $`(x-a)(x-b)=0`$ - - $`x = a, b`$ - -- - -**Note:** -- √x2 = ± x (There are 2 possible solutions) -- ```Distrubutive Property``` - This is opening the bracket. a(x+y) = ax+ay - -## Tips -- ```Absolute Values``` can have 2 solutions -- ```Quadratics``` can also have 2 solutions -- Make sure to do the reverse when moving things to the other side, meaning a positive on the ```left side``` becomes a negative on the ```right side``` - - - -# Unit 4: Measurement and Geometry -## Angle Theorems - -1. ```Transversal Parallel Line Theorems``` (TPT) - a. Alternate Angles are Equal ```(Z-Pattern)``` - b. Corresponding Angles Equal ```(F-Pattern)``` - c. Interior Angles add up to 180 ```(C-Pattern)``` - - - - -2. ```Supplementary Angle Triangle``` (SAT) - - When two angles add up to 180 degrees - - - - -3. ```Opposite Angle Theorem (OAT)``` (OAT) - - Two lines intersect, two angles form opposite. They have equal measures - - - - -4. ```Complementary Angle Theorem``` (CAT) - - The sum of two angles that add up to 90 degrees - - - - -5. ```Angle Sum of a Triangle Theorem``` (ASTT) - - The sum of the three interior angles of any triangle is 180 degrees - - - - -6. ```Exterior Angle Theorem``` (EAT) - - The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles - - - - -7. ``` Isosceles Triangle Theorem``` (ITT) - - The base angles in any isosceles triangle are equal - - - - -8. ```Sum of The Interior Angle of a Polygon``` - - 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 - - - - - -9. ```Exterior Angles of a Convex Polygon``` - - The sum of the exterior angle of any convex polygon is always ```360 degrees``` - - - - - -## Properties of Quadrilaterals -- Determine the shape using the properties of it - - |Figure|Properties| - |:-----|:---------| - |Scalene Triangle|no sides equal|Length of line segment| - |Isosceles Triangle| two sides equal|Length of line segment| - |Equilateral Triangle|All sides equal|Length of line segment| - |Right Angle Triangle|Two sides are perpendicular to each other| - |Parallelogram|Opposite sides are parallel and have equal length. Additionally, the diagonals bisect each other| - |Rectangle|Adjacent sides are perpendicular to each other. Furthermore, the diagonals bisect each other and are equal in length| - |Square|All sides are equal in length. The adjacent sides and diagonals are perpendicular. The adjacent sides are equal in length, so as the diagonals| - |Rhombus|Opposite sides are parallel and all sides are equal to each other, the diagonals are perpendicular| - |Trapezoid|There is one pair of opposite sides and they are parallel and unequal in length| - |Kite|The diagonals are perpendicular| - -## 2D Geometry Equations - |Shape|Formula|Picture| - |:----|:------|:------| - |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```: $`\frac{1}{3} b^2 h`$
```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) - -### 2D Objects - - |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
$`A = lw`$
$`A_{max} = (w)(w)`$
$`A_{max} = w^2`$|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 $`h = 2r`$
$`V = πr^2h`$
$`V_{max} = πr^2(2r)`$
$`V_{max} = 2πr^3`$|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: - 1. use ```CAPITAL LETTERS``` - 2. they have to be labeled in ```clockwise``` or ```counter-clockwise``` directions -- For a triangle, the side lengths are labeled in ```LOWERCASE LETTERS``` associated to the opposite side of the vertex - -- - -## Median -- Each median divides the triangle into 2 smaller triangles of equal area -- The centroid is exactly the way of each median from the vertex, or the way from the midpoint of the opposite side, or ```2:1``` ratio -- The three medians divide the triangle into ```6``` smaller triangles of equal area and ```3 pairs``` of congruent triangles - -- - -## Terms: -- ```Altitude``` The height of a triangle, a line segment through a vertex and perpendicular to the opposite side -- ```Orthocenter```: where all 3 altitudes of the triangle intersect - - -- ```Midpoint```: A point on a line where the length of either side of the point are equal -- ```Median```: A line segment joining the vertex to the midpoint of the opposite side -- ```Midsegment```: A line joining 2 midpoints of the 2 sides of a triangle -- ```Centroid```: The intersection of the 3 medians of a triangle - - - -## Proportionality theorem: -- The midsegment of a triangle is ```half``` the length of the opposite side and ```parallel``` to the opposite side -- Three midsegment of a triangle divide ```4 congruent``` triangles with the same area -- The Ratio of the outer triangle to the triangle created by the 3 midsegments is ```4 to 1``` -- - -## Tips -- Make sure to know your optimization formualas -- Read the word problems carefully, determine which formual to use -- Never **ASSUME**, be sure to **CALCULATE** as most of the time the drawings are **NOT ACCURATE** -- To find ```missing area```, take what you have, subtract what you don't want -- Don't be afraid to draw lines to help you solve the problem - - -# Unit 5: Analytical Geometry and Linear Relations -- ```Linear Relation```: A relation which a single straight line can be drawn through every data point and the first differences are constant -- ```Non - Linear Relation```: A single smooth curve can be drawn through every data point and the first differences are not constant - -## Slope and Equation of Line -- ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```the rate of change``` -- ```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``` -- Perpendicular slopes are negative reciprocals - -## Relations -- A relation can be described using - 1. Table of Values (see below) - 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. - -## Perpendicular Lines -- To find the perpendicular slope, you will need to find the slope point -- Formula: slope1 × slope2 = -1 -- 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 $`y`$. -- ```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. -- ```Collinear Points```: points that line on the same straight line - -## Variables -- ```Independent Variable```: A Variable in a relation which the values can be chosen or isn't affected by anything. -- ```Dependent Varaible```: A Variable in a relation which is **dependent** on the independent variable. - -## Statistics -- ```Interpolation```: Data **inside** the given data set range. -- ```Extrapolation```: Data **outside** the data set range. -- ```Line of Best Fit```: A line that goes through as many points as possible, and the points are the closest on either side of the line, -- and it represents the trend of a graph. -- ```Coefficient of Correlation```: The value that indicates the strength of two variables in a relation. 1 is the strongest and 0 is the weakest. -- ```Partial Variation```: A Variation that represents a relation in which one variable is a multiple of the other plus a costant term. - -## Time - Distance Graph -- Time is the independent variable and distance is the dependent variable -- You can't go backwards on the x-axis, as you can't go back in time -- Plot the points accordingly -- Draw the lines accordingly -- - -**Direction is always referring to:** - - 1. ```go towards home``` - 2. ```going away from home``` - 3. ```stop``` - -## Scatterplot and Line of Best Fit -- A scatterplot graph is there to show the relation between two variables in a table of values. -- A line of best fit is a straight line that describes the relation between two variables. -- 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. -- - -### How To Determine the Equation Of a Line of Best Fit - 1. Find two points **```ON```** the ```line of best fit``` - 2. Determine the ```slope``` using the two points - 3. Use ```point-slope form``` to find the equation of the ```line of best fit``` - -## Table of values -- To find first differences or any points on the line, you can use a ```table of values``` - - | y | x |First Difference| - |:--|:--|:---------------| - |-1|-2|.....| - |0|-1|(-1)-(-2) = 1| -|1|0|0 - (-1) = 1| -|2|1|1 - 0 = 1| -|3|2|2 - 1 = 1| -|4|3|3 - 2 = 1| - -## Tips -- Label your graph correctly, the scales/scaling and always the ```independent variable``` on the ```x-axis``` and the ```dependent variable``` on ```y-axis``` -- Draw your ```Line of Best Fit``` correctly -- Read the word problems carefully, and make sure you understand it when graphing things -- Sometimes its better not to draw the shape, as it might cloud your judgement (personal exprience) -- Label your lines - -# 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 - -# Credits -- Ryan Mark - He helped provide alot of information for me -- Ms Hung(Katie) - She helped me check over my study sheet, an amazing teacher! -- Magicalsoup - ME! diff --git a/Grade 9/Math/MPM1DZ/README.md b/Grade 9/Math/MPM1DZ/README.md new file mode 100644 index 0000000..9b84743 --- /dev/null +++ b/Grade 9/Math/MPM1DZ/README.md @@ -0,0 +1,56 @@ +# Math Study Sheet!!!! +- Due do performance, issues, I decieded to split the study sheet into the 6 units, hope that doesn't hinder anyone. + +# Exam Detail +|Unit|Marks| +|:---|:----| +|Unit 1|10| +|Unit 2|10| +|Unit 3|9| +|Unit 4|11| +|Unit 5|11| +|Unit 6|8| +|Forms|4| +|Total|63| + +|Section|Marks| + |:------|:----| + |Knowledge|21| + |Application|23| + |Thinking|12| + |Communication|3| + |Forms|4| + + |Part|Question| + |:---|:-------| + |A|9 multiple choice| + |B|10 Short Answer -->
- 7 Knowledge questions
- 3 Application Questions| + |C|10 Open Response -->
- 10 Knowledge Questions
- 5 Application Questions
- 3 Thinking Questions
- 1 Communication Question| + + +# 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 + +# Credits +- Ryan Mark - He helped provide alot of information for me +- Ms Hung(Katie) - She helped me check over my study sheet, an amazing teacher! +- Magicalsoup - ME!