Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Simple Equations and Word Problems Prepared by: Richard Mitchell Humber College C3

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-Solving First-Degree Equations

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-STRATEGY-Page 86 Eliminate Fractions Eliminate Fractions: Multiply both sides of the equation by the lowest common denominator. Remove Parenthesis Remove Parenthesis: Brackets are multiplied away. Collect x Terms Collect x Terms: Move all x terms to one side and all other terms to other side. Combine Like Terms Combine Like Terms: Always simplify. Remove Coefficient of x Remove Coefficient of x: Divide both sides by coefficient. Check Answer Check Answer: Be sure to substitute the answer back into the original equation.

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 4-Page 84 Solve the equation for x: 3x = 8 + 2x 3x – 2x = 8 + 2x – 2x x = x = 8 CHECK: 3(8) = 8 + 2(8) 24 = = 24 (checks) ANS: x = 8

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 7-Page 85 Solve the equation for x: 3(3x + 1) – 6 = 5(x – 2) x + 3 – 6 = 5x – x – 3 = 5x + 5 9x – 5x = x = 8 x = 2 CHECK: 3[3(2)+1] - 6 = 5[(2)-2] [6 + 1] – 6 = 5[2 – 2] [7] – 6 = 5[0] – 6 = = 15 (checks) ANS: x = 2

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 8-Page 86 Solve the equation for x: x – 6 = 15 x = x = 21 CHECK: 7 – 2 = 5 5 = 5 (checks) ANS: x = 21

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 9-Page 86 Solve the equation for x: 3x + 18 = 2x 3x – 2x = -18 x = -18 CHECK: = = -6 (checks) ANS: x = -18

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 10-Page 87 Solve the equation for x: 3(3x - 5) = 4(x - 1) + 6(4) 9x – 15 = 4x x – 4x = x = 35 x = 7 ANS: x = 7 (con’t)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 10-Page 87 Solve the equation for x: CHECK: ANS: x = 7

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 11-Page 87 Solve the following for x: 3(x + 2)(2 – x) = 3(x – 2)(x – 3) + 2x(4 - 3x) 3[2x – x – 2x] = 3[x 2 – 3x – 2x + 6] + 8x – 6x 2. 3[– x 2 + 4] = 3[x 2 - 5x + 6] + 8x – 6x 2 -3x = 3x 2 – 15x x – 6x 2 12 = -15x x 15x – 8x = 18 – 12 7x = 6 (x – 2)[2x 2 – x – 3]

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 11-Page 88 Solve the following for x: 3(x + 2)(2 – x) = 3(x – 2)(x – 3) + 2x(4 - 3x)CHECK: 3(6/7 + 2)(2 – 6/7) = 3(6/7 – 2)(6/7 – 3) + 2(6/7)[4 – 3(6/7)] 3(20/7)(8/7) = 3(-8/7)(-15/7) + 12/7[10/7] 9 39/49 = 7 17/ / /49 = 9 39/49 (checks)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.1-EXAMPLE 15-Page 89 Solve the equation for x: 3x + b = 5 3x = 5 - b CHECK:

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.2-Solving Word Problems

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.2-STRATEGY-Pages 90 to 94 Verbal Statements Picture the Problem Picture the Problem: Use a detailed sketch to help visualize the problem. Understand the Words Understand the Words: Look up meanings of unfamiliar words (eg. Difference, quotient, consecutive etc). Identify the Unknown Identify the Unknown: Determine exactly what is to be found (eg.find, calculate, how much etc). Define Other Unknowns Define Other Unknowns: Define additional unknowns in terms of the original unknown if possible. Estimate the Answer Estimate the Answer: Estimate or guess the answer to see if the result is reasonable. Write and Solve the Equation Write and Solve the Equation: Use an equation or formula to relate the unknown quantity to the given quantities. Always Check Your ANSWER Addition Sum, total, and, plus, increased by, more than, added to, risesSubtraction Difference, minus, less, remains, is subtracted from, decreases by, diminished byMultiplication Times, product, of, multiplied by, triple, twiceDivision Divided, dividend, remainder, quotient, split into, equal amounts

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.2-EXAMPLE 26-Page 94 Let x = the number Solve for x Check Your ANSWER Twice the number 11 is equal to 22. Twice this number increased by nine is equal to 31. Triple the number 11 is equal to 33. Two less than triple this number is also equal to 31. Thus, our number 11 checks.

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.2-EXAMPLE 28-Page 95 Let x = number of employees on the night shift Check Your ANSWER If x is 20, then there are 20 employees on the night shift. Three times as many would equal 60 on the day shift. Two more than the night shift would equal 22 on the swing shift. Let 3x = number of employees on the day shift Let (x + 2) = number of employees on the swing shift Equation

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.3-Financial Problems

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.3-EXAMPLE 30-Page 97 AMOUNTofInvestmentRATEofInterestINTERESTAccountA AccountB TOTALS x = x ( – x) 0.06 (x) 0.08 ( – x) Equation

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.4-Mixture Problems

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.4-EXAMPLE 34-Page 99 AMOUNTofSteelPERCENTofNickle AMOUNT of NICKLE MixtureA MixtureB TOTALS x= (3.25) x (3.25 – x) (x) (3.25 – x) Equation

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.5-Statics Problems

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 3.5-EXAMPLE 36-Page 102 ESTIMATE ESTIMATE: If the 525 N load were at the mid-span, the two reactions would have equal values of ½(525) or N. Since the left reaction (315 N) is greater than that, we deduce that the load is to the left of the mid-span and that the reaction at the right will be less than N. Taking moments about p, we set the moments that tend to turn the bar in a clockwise (CW) direction equal to the moments that tend to turn the bar in a counter-clockwise (CCW) direction. (By Eq.A15)

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