the best algebra notes

​1.​​Linear​​Equations​​–​​The​​Foundation​​(must​​be​​automatic)​

​Golden​​5-step​​method​​(use​​every​​single​​time):​

​ .​
1 ​ lear​​any​​parentheses​​(distribute)​
C
​2.​ ​Move​​all​​variable​​terms​​to​​left,​​constants​​to​​right​
​3.​ ​Combine​​like​​terms​​on​​each​​side​
​4.​ ​Divide​​/​​multiply​​last​​(coefficient​​of​​variable​​→​​1)​
​5.​ ​Verify​​(plug​​answer​​back​​–​​30​​seconds​​but​​saves​​marks)​

​Most​​common​​traps​​examined​​:​

​​ –
● ​ (3x​​–​​5)​​=​​–3x​​+​​5​​←​​sign​​change​​mistake​
​●​ ​5​​–​​2(x​​+​​3)​​=​​5​​–​​2x​​–​​6​​=​​–2x​​–​​1​​←​​very​​frequent​​error​
​●​ ​Fractions:​​multiply​​through​​by​​LCD​​before​​moving​​terms​

​Literal​​equations​​(very​​frequent​​3–4​​mark​​question)​

​ xample:​​Make​​r​​the​​subject​
E
​A​​=​​P(1​​+​​rt)​
​A​​=​​P​​+​​Prt​
​A​​–​​P​​=​​Prt​
​(A​​–​​P)/P​​=​​rt​

​r​​=​​(A​​–​​P)/(Pt)​

​2.​​Inequalities​​–​​The​​Sign​​Flip​​Rule​​Is​​Everything​

​Four​​danger​​zones​​examined​​every​​year​​:​

​ .​
1 ​ ultiplying​​/​​dividing​​by​​negative​​number​​→​​flip​
M
​2.​ ​–x​​>​​5​​→​​multiply​​both​​sides​​by​​–1​​→​​x​​<​​–5​
​3.​ ​Fraction​​inequalities:​​–2/x​​≥​​3​​(never​​multiply​​/​​divide​​by​​variable​​unless​​you​​know​​sign)​
​4.​ ​|x​​–​​3|​​≤​​7​​→​​–7​​≤​​x​​–​​3​​≤​​7​​→​​–4​​≤​​x​​≤​​10​

​Compound​​inequality​​layout​​students​​lose​​marks​​on​​:​

-​ 3​​≤​​2x​​+​​5​​<​​11​
​–8​​≤​​2x​​<​​6​

​–4​​≤​​x​​<​​3​ ​←​​final​​answer​​must​​look​​like​​this​

​3.​​Coordinate​​Geometry​​/​​Straight​​Lines​​–​​Huge​​mark​​earners​

, ​Three​​forms​​–​​know​​when​​each​​is​​easiest​

​Situation​ ​Best​​form​ ​Why​

​Given​​slope​​&​​y-intercept​ ​y​​=​​mx​​+​​b​ ​Instant​

​Given​​slope​​&​​one​​point​ ​y​​–​​y₁​​=​​m(x​​–​​x₁)​ ​Direct​

​Given​​two​​points​ ​ irst​​find​​m,​​then​​use​
F ​Reliable​
​point-slope​

​Want​​Ax​​+​​By​​=​​C​​form​ ​Rearrange​​last​ ​ ometimes​
S
​asked​

​Parallel​​&​​perpendicular​​–​​automatic​​recall​

​​ P
● ​ arallel​​→​​m₁​​=​​m₂​
​●​ ​Perpendicular​​→​​m₁​​×​​m₂​​=​​–1​​(or​​one​​is​​vertical​​&​​one​​horizontal)​

​Midpoint​​formula​​(very​​common​​2-mark​​question)​

​((x₁+x₂)/2​​,​​(y₁+y₂)/2)​

​Distance​​formula​​(often​​hidden​​in​​proving​​shapes)​

​√[(x₂–x₁)²​​+​​(y₂–y₁)²]​

​4.​​Simultaneous​​Equations​​–​​Elimination​​vs​​Substitution​

​Decision​​tree​​for​​exam​​speed​

​​ O
● ​ ne​​equation​​already​​has​​y​​=​​or​​x​​=​​→​​Substitution​
​●​ ​Coefficients​​match​​or​​easy​​to​​make​​match​​→​​Elimination​
​●​ ​Messy​​decimals/fractions​​→​​Graphing​​(if​​allowed)​​or​​substitution​

​Elimination​​–​​make​​coefficients​​equal​​trick​

​2x​​+​​3y​​=​​8​

​5x​​–​​2y​​=​​11​

​ 2​​on​​first​​→​​4x​​+​​6y​​=​​16​
×
​×3​​on​​second​​→​​15x​​–​​6y​​=​​33​

​Add​​→​​19x​​=​​49​​→​​x​​=​​49/19​

​5.​​Quadratic​​Equations​​–​​The​​Heart​​of​​Algebra​​1​​&​​2​​Exams​