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2.1: Linear Equations

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.

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2.1: Linear Equations

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  1. 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. 3 + 6 = 9 x + 3 = 13 x + 4 = 20 – 3x

  2. +1 +1 -x -x -8 -8 2 2 -6 -6 3 3 x = ½ x = -5/3 x = -2 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 = 2 2x = 1 -6x = 10 3x = -6

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

  4. 100 Clearing Decimals .06x + 0.09(15 – x) = 0.07(15) 6x + 9(15 – x) = 7(15) 6x + 135 –9x = 105 -3x + 135 = 105 -135 -135 -3x = -30 -3 -3 x = 10

  5. 3 Types of Linear Equations 5x - 9 = 4(x – 3) 5x – 9 = 4x – 12 -4x -4x ---------------------- X – 9 = -12 + 9 + 9 ----------------------- X = -3 5x – 15 = 5(x – 3) 5x – 15 = 5x – 15 + 15 + 15 ---------------------------- 5x = 5x -5x -5x ---------------------------- 0 = 0 5x – 15 = 5(x – 4) 5x – 15 = 5x – 20 + 15 + 15 ---------------------------- 5x = 5x -5 -5x -5x ---------------------------- 0 = -5 Infinite Solutions (All Real Numbers) IDENTITY NO Solutions (Null Set : O ) CONTRADICTION 1 Solution CONDITIONAL

  6. 2.2-2.3 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 = RT A = LW I = PRT D – distance A – Arearectangle I - Interest R – rate L – Length P – Principal ($$ borrowed/invested) T – time W – 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 -2L P – 2L = 2W 2 2 Solve for W: Solve for B P = 2(L + W) N = A + B 2

  7. Change a Percent to a Decimal  Move the decimal point two places to the left 45% =.45 5% =.05120% =1.23.2% =.032 500% = 5 Percentages 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% 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 What is 25% of 70? X = .25 • 70 X = 17.5 16 is what percent of 50 16 = 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

  8. Word Problems/Applications • Tips on word problems: • 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

  9. #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. Classic Problems #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?

  10. 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

  11. 2.4-2.5 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 less x such that x is greater x is greater than –4 than or equal to 2 than –2 and less than or equal to 7 Interval (-4, ) (-, 2] (-2, 7] Notation Graph -4 0 2 7 -2 Question: How would you write the set of all real numbers? (-, ) or R

  12. 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

  13. Three-Part Inequality -3 < 2x + 1 < 3 Set Notation: {x | -2 < x < 1} -1 -1 -1 -4 < 2x < 2Interval Notation: (-2, 1] 2 2 2 Graph: -2 < x < 1 0 -2 1 An Inequality Word Problem: (P. 107) : Average Test Score Martha has scores of 88, 86, and 90 on her 1st 3 tests. An average score of 90 Will earn her an A in the course. What does she need on her 4th test to have An A average? 88 + 86 + 90 + x 90 4

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

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