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MPM1D Date: ________________ 4.1 Solve Simple Equations To maintain equilibrium in an equation you must perform the same operation on both sides of the equation. To ISOLATE a variable you need to perform the opposite operation. Starting backwards from BEDMAS... SAMDEB Undo additions & subtractions first then multiplications and divisions. What is the opposite operation? addition-------> subtraction-----> multiplication----> division--------> exponent 2------> square root-----> Ex: Solve a) m + 3 = 4 Communication: Keep equal signs aligned, only one per line MPM1D Date: ________________ b) x - 2 = 6 c) 4x = 20 d) x 12 3 2 e) x 36 MPM1D Date: ________________ Always remove the constants before removing the coefficients! f) 2x - 11 = 27 Check your solution to example 2 f) Checking a solution Substitute the root into the right and left side of the equation. Both sides must be equal. Communication: You must separate the Left Side (LS)from the Right Side (RS) of the equation. g) 5 x 3 2 MPM1D h) 3 x Date: ________________ 2 4 5 i) 5m 4 2 3 Ex 3: At a computer store, packages of DVDs sell for $15 each. a) Write an equation to model the number of packages of DVDs bought. b) Given that one customer buys $120 worth of DVDs use your equation to solve for the number of packages of DVDs. Let represent the number of packages of DVDs MPM1D Date: ________________ 4.2 Solve Multi-Step Equations Recall: To solve equations "undo" additions & subtractions first, then multiplications & divisions. To solve equations with variables on both sides, use inverse operations to group the variable terms on one side of the equation. Ex. 1 Solve a) 3x - 7 = 8x + 8 b) 2x + 8x - 4 = 6x + 10 - 8 c) 5(x + 4) = 3x + 14 MPM1D d) 7 - ( 4m + 3 ) = -3 (m + 2) - (2m + 3) Ex. 2: Check your solution to Ex. 1 d) Date: ________________ MPM1D Date: ________________ Ex. 3: Two or more angles are supplementary if their sum is 180 degrees. An angle is 4 times the value of its supplement. Set up and solve an equation to find the measures of the two angles. Ex. 4: The square and equilateral triangle shown have the same perimeters. What are the dimensions of each figure? MPM1D Date: ________________ 4.3 Solve Equations Involving Fractions Eliminate fractions by multiplying every term by the common denominator Ex. 1 Solve a) x 2 6 b) 1 a 6 7 2 c) x 1 7 3 Communication: Keep equal signs aligned, only one per line MPM1D d) m m 2 2 3 e) 3 2m m 4 3 f) x 3x 1 x 3 2 4 6 Date: ________________ MPM1D g) Date: ________________ 2 1 (x 5) (x 2) 2 3 4 Ex. 2: A triangular backyard has dimensions as shown. a) Write an algebraic expression for the perimeter. b) Determine the dimensions of the yard if the perimeter is 60 m. MPM1D Date: ________________ 4.4 Modelling with Formulas Formulas can be rearranged to isolate different variables. Isolate the variable that is requested! Use the same process used to solve equations. Ex. 1 Solve for the variable indicated. a) y mx b solve for b b) y mx b solve for m 2 c) V r h solve for r solve for h 2 d) A r MPM1D e) 4 x 3y 15 solve for y Date: ________________ f) 2 x 8y 10 Ex. 2 The formula for the area of a trapezoid is A solve for y h (a b) 2 Solve for b then find the length of side b if the area is 84 cm2 , the height is 8 cm, and the length of side a is 12 cm. MPM1D Date: ________________ 4.5A Modelling with Algebra 1. State the correct operation twice ___________ half ___________ difference ___________ warmer ___________ product ___________ quotient ___________ fewer ___________ is ___________ times ___________ older ___________ reduced ___________ younger ___________ are ___________ more ___________ triple ___________ quarter ___________ taken from ___________ gives us ___________ 2. If n represents a number, write an algebraic expression using numbers and symbols for each of the following statements. a) Three times a number _______________________ b) A number increased by one _______________________ c) A number decreased by five _______________________ d) Four more than twice a number _______________________ e) Half a number _______________________ f) Double a number reduced by six _______________________ MPM1D Date: ________________ Number Problems 1) A number divided by 2, increased by 6 is 11. Find the number. 2) One number is 2 more than 3 times another number. If the sum of the numbers is 14, what are the numbers? MPM1D Date: ________________ 3) Double the square of a number is increased by 2 resulting in 52. What is the number? 4) Find two consecutive numbers with a sum of 149. Consecutive Numbers: x, x + 1, x + 2, ... 5) The sum of two numbers is 34. The difference between the two numbers is 4. Find the two numbers. MPM1D Date: ________________ 4.5 B Modelling with Algebra Measurement Porblems Ex. 1 The sides of a triangle are 3 consecutive whole numbers. The perimeter of the triangle is 48 cm. How long is each side? Let represent the smallest number. Ex. 2 The length of a rectangle is 3 m greater than the width. The perimeter is 26 m. What are the dimensions of the rectangle? MPM1D Date: ________________ Money Problems Ex. 3 Heather earned $3 more than double the amount Oliver earned. The difference of their earnings was $15. How much did each person earn? Ex. 4 A parking meter contains $27.05 in quarters and dimes. There are 146 coins. How many quarters are there? MPM1D Date: ________________ Other Word Problems Ex. 5 Farren's mother is 4 years older than twice Farren's age. The difference of their ages is 22 years. Find their ages. Ex. 6 Two cars left the same highway restaurant at the same time but drove in opposite directions. One travelled at 65 km/h, the other at 55 km/h. After how long were they 600 km apart?