2.1: Linear Equations Algebra Representing real-world situations with mathematical expressions & statements Solving real-world and/or mathematical problems involving unknown quantities SEARCHING FOR THE UNKNOWN: A VARIABLE A variable is a symbol, usually a letter (like X, Y, T, P) that is used to represent an unknown number. If Sam ate many tacos and we don’t know how many, we might say Sam ate X tacos. Tomorrow if Sam eats 3 more than he did today, we could say Sam eats X+3 tacos tomorrow. X and X+3 are algebraic expressions representing the number of tacos eaten. If Sam ate 10 tacos today, then he will eat X + 3 = (10) + 3 = 13 tomorrow. If Sam ate 4 tacos today, then he will eat (4) + 3 = 7 tacos tomorrow. An equation is the equality of two algebraic expressions = 9x + 3 = 13x + 4 = 20 – 3x

Linear Equations in 1 Variable A linear equation in one variable is an equation that can be written as: Ax + B = C where A, B, C  R, A  0 To solve a linear equation, find all variable values that make the equation true. These values are called the solution set. Steps to solving symbolically: Step 1: Locate both sides of the equation (separated by the ‘=‘ sign) Step 2: Clear any fractions or decimals Step 2: Simplify each side separately : Use distribution & combine like terms Step 3: Move the ‘variable terms’ to one side and ‘number terms’ to the other Step4: Reverse/Inverse what is happening to X until you have X = __ Step5: Check your answer by plugging X back in Examples: 2x –1 = 0 -5x = 10 + x 3x + 8 = x -x x = 1 -6x = 10 3x = x = ½ x = -5/3 x = -2

Distribution & Clearing Fractions 2 (x – 1) = 4 – ½ (4 + x) 1 (2x - 3) –1 x = [Multiply by Common Denominator] 6 2 (2x –3) –3 x = -12 4x x = -12 x - 6 = = +6 x = -6 2x –2 = ½ x 2x – 2 = 2 - ½ x x = 4 – ½ x + ½ x + ½ x 2 ½ x = 4 (5/2) x = 4 x = 8/5 (2/5)

Clearing Decimals.06x (15 – x) = 0.07(15) 100 6x + 9(15 – x) = 7(15) 6x –9x = x = x = x = 10

3 Types of Linear Equations 5x - 9 = 4(x – 3) 5x – 9 = 4x – 12 -4x X – 9 = X = -3 5x – 15 = 5(x – 3) 5x – 15 = 5x – x = 5x -5x -5x = 0 5x – 15 = 5(x – 4) 5x – 15 = 5x – x = 5x -5 -5x -5x = -5 1 Solution CONDITIONAL Infinite Solutions (All Real Numbers) IDENTITY NO Solutions (Null Set : O ) CONTRADICTION

Formulas & Equations A formula is an equation that can calculate one quantity by using a known value of another quantity. Formulas usually involve real-world applications. D = RTA = LWI = PRT D – distanceA – Area rectangle I - Interest R – rateL – LengthP – Principal ($$ borrowed/invested) T – timeW – Width T – Time (years) If Anna travels 50mph for 15 hours, how far did she travel? D = RT D = (50)(15) = 750 miles Formulas can be solved for a specific variable P = 2L + 2W (solve for W) -2L P – 2L = 2W 2 2 Solve for W: Solve for B P = 2(L + W) N = A + B 2

Percentages Change a Percent to a Decimal  Move the decimal point two places to the left 45% =.45 5% = % = % = % = 5 Change a Decimal Number to a Percent  Move the decimal point two places to the right.45 = 45%.05 = 5% 1.2= 120%.032= 3.2% 5 = 500% What is 25% of 70? X = X = is what percent of = X 50 X = 16/50 =.32 = 32% A man weighed 150 lbs last year. This year the same man weighs 175 lbs. What was the percent increase from last year to this year.? Difference = 25 =.167 = 16.7% increase Original 150 A class has 50 students. 32 are males. What is the percent of males in the class? Partial amount = percent 32 =.64 = 64% Whole amount 50

Word Problems/Applications Tips on word problems: 1.Read the problem through once entirely, then go back and read it again noting the important information. You may have to read it more times too as you work the problem & you may wish to organize your thoughts with pictures or charts. 2. Assign variables for unknown quantities & anything you need to find. 3. Write equation(s) related to the problem using your variables. (Translate words/sentences in the problem into an algebraic equation) 4. Solve the equation & check your solution to see if it is reasonable. Examples Find the number: Twice a number, decreased by 3 is 42 : 2x –3 = 42 The quotient of a number and 4 plus the number is 10: x + x = 10 4

Classic Problems #1 Geometric Dimensions:The length of a rectangle is 1cm more than twice the width. The perimeter of the rectangle is 110 cm. Find the length and the width of the rectangle. #2 Percent Interest: Mark had $40,000 to invest. He puts part of the money in the bank earning 4% interest and the rest in stocks paying 6% interest for an annual income of $2040. Find the amounts in the bank and in stock. #3 Acid Mixture: a chemist mixes 8 L of 40% acid solution with some Unknown quantity of 70% solution to get a 50% solution. How much 70% Solution is used? #4 Coins: A bill is $5.65. The cashier received 25 coins (all nickles & quarters). Howmany of each coin did the cashier receive?

Investment Formula/Problem (P. 70 – Example 4) Karen Estes just received an inheritance of $10000 And plans to place all money in a savings account That pays 5% compounded quarterly to help her son Go to college in 3 years. How much money will be In the account in 3 years? Use the formula: A = P(1 + r/n) nt A = amount in account after t years P = principal or amount invested T = time in years R = annual rate of interest N = number of times compounded per year

Inequality Set & Interval Notation Set Builder Notation {1,5,6}{ }  {6} {x | x > -4}{x | x < 2}{x | -2 < x < 7} x such that x such that x is lessx such that x is greater x is greater than –4 than or equal to 2than –2 and less than or equal to 7 Interval (-4,  ) (- , 2] (-2, 7] Notation Graph Question: How would you write the set of all real numbers? (- ,  ) or R

Inequality Example StatementReason 7x + 15 > 13x + 51 [Given Equation] -6x + 15 > 51 [-13x from both sides] -6x > 36 [-15 from both sides] x < -6[Divide by –6, so must ‘flip’ the inequality sign Set Notation: {x | x < -6} Interval Notation: (- , -6] Graph: -6

Three-Part Inequality -3 < 2x + 1 < 3 Set Notation: {x | -2 < x < 1} < 2x < 2 Interval Notation: (-2, 1] Graph: -2 < x < An Inequality Word Problem: (P. 107) : Average Test Score Martha has scores of 88, 86, and 90 on her 1 st 3 tests. An average score of 90 Will earn her an A in the course. What does she need on her 4 th test to have An A average? x  90 4

Set Operations and Compound Inequalities Union (  ) – “OR” A  B = {x | x  A or x  B} -4x + 1  9 or 5X+ 3  12 X  -2 or X  -3 Intersection (  ) – “AND” A  B = {x | x  A and x  B} X+ 1  9 and X – 2  3 X  8 and X  5 Set Notation: {x | X  8 and X  5} Interval Notation: (- , 8]  [5,  ) 0 58 [ ] Set Notation: {x | X  -2 or X  -3} Interval Notation: (- , -2]  (- , -3] -2