From 27eaf73b9ff45f556609f2af13729231ca2ef7d6 Mon Sep 17 00:00:00 2001 From: Soup <39444003+magicalsoup@users.noreply.github.com> Date: Tue, 15 Jan 2019 08:12:07 -0500 Subject: [PATCH] Create Math.md --- Grade 9/Math.md | 585 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 585 insertions(+) create mode 100644 Grade 9/Math.md diff --git a/Grade 9/Math.md b/Grade 9/Math.md new file mode 100644 index 0000000..651c255 --- /dev/null +++ b/Grade 9/Math.md @@ -0,0 +1,585 @@ +# Math Study Sheet!!!! + +# Essential Skills (1) +> ## 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``` +>> ```W``` represents **Whole Numbers** (W = {x | x > 0, x ∈ Z}) +>> ```N``` represents **Natural Numbers** (N = {x | x ≥ 0, x ∈ Z}) +>> ```Z``` represents **Integers** (Z = {x | -∞ ≤ x ≥ ∞, x ∈ Z}) +>> ```Q``` represents **Rational Numbers (Q = {a⁄b |a, b ∈ Z, b ≠ 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 +>> let a be the adjecant and b be the opposite, and c be the hypotenuse. +>> a2 + b2 = c2 + +>> + +> ## Operations with Rationals +>> Q = { a⁄b | a, b ∈ Z, b ≠ 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: +>>> = 3⁄4 ÷ 2⁄12 [Reduce to lowest terms] +>>> = 3⁄4 ÷ 1⁄7 [Multiply by reciprocal] +>>> = 3⁄4 × 7 +>>> = 21⁄4 [Leave as an improper fraction] + +>> ### Shortcut for multiplying fractions +>>> cross divide to keep your numbers small +>>> Example: +>>> 2⁄4 × 4⁄6 +>>> 1⁄1 × 1⁄3 +>>> = 1⁄3 + +>> ## 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|(a⁄b)n = an⁄bn|(2⁄3)4 = 24⁄(34)| +>>> |Zero as Exponents|a0 = 1|210 = 1| +>>> |Negative Exponents|a-m = 1⁄am|1-10 = 1⁄(110) or 1⁄1| +>>> |Rational Exponents|an/m = (m√a)n|165/4 = (4√16)5 = 25| + +>>> **Note:** +>>> Standard --> 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 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. +>>> When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used + +>>> + +>> ## 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 + + +# Polyomials (2) +> ## 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 = 5)``` +>> 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 = a2+2ab+b2 or (a-b)2 = a2-2ab+b| +>> |Difference with Squares|a2-b2 = (a+b)(a-b)| +>> |Simple Trinomials|x2+6x-7 = (x+7)(x-1)| +>> |Complex Trinomials|2x2-21x-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 + +# Solving Equations and Inequailties (3) +> ## 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``` +> ## Venn Diagrams + +>> + +>> ```Set```: a collection of elements, O (the circle) +>> ```Universal Set```: This is a collection of all the elements that you are interested in. Use ```{}``` bracket to write the set inside the rectangle +>> ```Union```: ∪ +>> if 2 sets have union, all the elements belong to any of the set. This is known as ```or``` +>> ```Intersection```: ∩ +>> if 2 sets have an intersection, they are elements belonging to both sets. This is known as ```and``` +>> ```Set Notation```: +>> A notation that represents the collection of numbers. It is written in this form x = {x|x∈R} +>> ```Absolute Value``` +>>> 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 +>>> if x > 0, | x | = x +>>> if x == 0, | x | = 0 +>>> if x < 0, | x | = -x + +> ## 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 +>> ## 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)``` + +>>> + +> ## 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 = b2 - 4ac** + +> ## 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``` + + + +# Measurement and Geometry (4) +> ## 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 + +> + +> ## 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```: bh/2
```Perimeter```: a+b+c|| +>> |Circle|```Area```: πr2
```Circumference```: 2πr or πd|| +>> |Trapezoid|```Area```: (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|| + + +> ## 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
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| + + +>> ### 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 = 2πr2|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| + +> ## 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 2⁄3 they way of each median from the vertex, or 1⁄3 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 + + +# Analytical Geometry and Linear Relations (5) +> ```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 ```change in y / change in x``` +>> ```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)** +>> The slope of a vertical lines is undefined +>> The sloope 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 Bisector +>> To find the perpendicular bisector, you will need to fidn the slope and midpoint +>> Formula: slope1 × slope2 = -1 +>> Notation: m⊥ +>> + + +> ## 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 +>> + +> ## 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. +>> + +> ## 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|1| +>>|0|-1|1| +>>|1|0|1| +>>|2|1|1| +>>|3|2|1| +>>|4|3|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 + +# System of Equations (6) + +> ## 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 +>>> + +> # 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 ≥ mx + b| +>> |Shade the region ```below``` the line|y < mx + b| y ≤ mx + b| + +>> ## If + +>> |x > a
x ≥ a| +>> |:------------------| +>> shade the region on the **right** + +>> ## If + +>> |x < a
x ≤ 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 + +> ## 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 +> Magicalsoup - ME!