MATH - Q1 PDF - Linear Equations & Inequalities

Lesson 1 - Linear Equations Linear Equations Linear Equations Have 1 variable Always has the equal sign/= Ax + B = C Conditions Only 1 variable Highest value of the variable is 1 A, B, C are real numbers A = 0; Ax is the coefficient Solution se...

Lesson 1 - Linear Equations Linear Equations Linear Equations Have 1 variable Always has the equal sign/= Ax + B = C Conditions Only 1 variable Highest value of the variable is 1 A, B, C are real numbers A = 0; Ax is the coefficient Solution set A set of all replacements which makes the equation true Types of Solution: Conditional True for only a particular value Ex. - 4x - 10 = 2x - 4x - 2x - 10 = 2x - 2x - [2x = 10] 2/1 - x=5 Contradiction No solution; Null/empty set Also known as inconsistent Ex. - 2(x - 1) = 2x - 3 - 2x - 2x = -3 + 1 - 0 = -1 : No solution Identity Any value is equal to itself; 0 = 0 Ex. - 4x - 4x = -20 + 20 - 0 = 0 : Identity Equation Properties of Equality: Reflexivity (Reflexive Property) Any real number is equal to itself Symmetry (Symmetric Property) If a = b; b = a Transitivity (Transitive Property) If a = c, b = c, then a = b Addition Property of Equality / Multiplication Property of Equality If a = b, then a + c = b + c If a = b, then a(c) = b(c) Any number added/multiplied to both sides of the equation does not affect the nature of it. Translating English Expression into an Algebraic Expression Addition Sum of, plus, added to, more than, increased by Subtraction Difference of, minus, subtracted from, less than, decreased by Multiplication Product of, times, (twice, triple etc) Division Quotient of, divided by, divided into, ratio of Exponents Squared, cubed, squared of, raised to the nth power Roots Square root of n Lesson 2 - Problem Solving Consecutive Integers Numbers that follow an order Ex. 3 numbers added equals to 6 - X = 1st - X + 1 = 2nd - X + 2 = 3rd - X+X+1+X+2=6 - 3x + 3 = 6 - X=1 Ex. 2 3 consecutive EVEN numbers equal to 24 - X = 1st - X + 2 = 2nd - X + 4 = 3rd - X + X + 2 + X + 4 = 24 - 3x + 6 = 24 - X=6 Age Problems Understand Translate Solve In x years (add) / x years ago (subtract) Ex. Matthew is 3 years older than Gail. How old are they if the sum of their ages are equal to 25 in 5 years? Age Present In 5 years Gail x X+5 Matthew X+3 X+3+5 - X + 5 + x + 3 + 5 = 27 - 2x + 13 = 27 - 2x = 14 - X=7 Percentage Problems P is the Percentage R is the Rate B is the Base Percentage Base Rate Interest Principal Rate of Interest Discount Original Price Rate of Discount Commission Sales Rate of allotment Tax Taxable amount Tax rate Rate of commissions Cancellation is done rectangularly Formula: - P= R B 100 Percent Always out of 100 1% means one part out of a hundred 25% = ¼; 50% = ½ 100% represents the whole amount Discount Amount reduced from the original price  Sale Price = Original Price - Discount Mark up Amount added to the original price Interest Amount paid for the use of money I = P (R) T I = New Selling price - Old selling price Compound Interest Amount of times interest was taken FV (Future Value) = Principal (1 + Interest Rate)^no. of times interest was compounded/time Annual = n Semiannual = n x 2 Quarterly = n x 4 Commission Percentage of the Total Sales for a business transaction C = Total Sales x Rate of Commission Percent Change Decrease/Increase of a product Expressed as a percentage of the original amount Increase - (Increase / Original Value) 100 Decrease - (Decrease / Original Value) 100 Note: If the remainder is less than half of the divisor, keep as is. Iff it is more than half, add 1 to the last digit of the quotient R >½0 Investment.Denotes the simple interest I = P (R) Lesson 3 - Inequalities Linear Inequality A given algebraic relation is either greater than or less than - < is the less than symbol - > greater than  - < less than or equal to - > greater than or equal to Trichotomy If a and b are real numbers, one of the following is true - A>B - A=B - A