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Chapter 1: Tools of Algebra 1-3: Solving Equations Essential Question: What is the procedure to solve an equation for a variable?

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1-3: Solving Equations The solution of an equation is a number that can be used in place of the variable that makes the equation true. You can manipulate equations to help find a solution, so long as you do the same thing to both sides of the equation. Addition PropertyIfa = b, then a + c = b + c Subtraction PropertyIf a = b, then a – c = b – c Multiplication PropertyIf a = b, then ac = bc Division PropertyIf a = b, then a / c = b / c

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1-3: Solving Equations Solve 13y + 48 = 8y – 47

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1-3: Solving Equations Solve 13y + 48 = 8y – 47 13y + 48 = 8y – 47 – 48 – 48(subtract 48 from both sides) 13y = 8y – 95

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1-3: Solving Equations Solve 13y + 48 = 8y – 47 13y + 48 = 8y – 47 – 48 – 48(subtract 48 from both sides) 13y = 8y – 95 –8y –8y(subtract 8y from both sides) 5y = – 95

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1-3: Solving Equations Solve 13y + 48 = 8y – 47 13y + 48 = 8y – 47 – 48 – 48(subtract 48 from both sides) 13y = 8y – 95 –8y –8y(subtract 8y from both sides) 5y = – 95 5 5(divide both sides by 5) y = -19

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1-3: Solving Equations Solve 3x – 7(2x – 13) = 3(-2x + 9)

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1-3: Solving Equations Solve 3x – 7(2x – 13) = 3(-2x + 9) 3x – 7(2x – 13) = 3(-2x + 9) 3x – 14x + 91 = -6x + 27(distribute)

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1-3: Solving Equations Solve 3x – 7(2x – 13) = 3(-2x + 9) 3x – 7(2x – 13) = 3(-2x + 9) 3x – 14x + 91 = -6x + 27(distribute) -11x + 91 = -6x + 27(combine like terms)

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1-3: Solving Equations Solve 3x – 7(2x – 13) = 3(-2x + 9) 3x – 7(2x – 13) = 3(-2x + 9) 3x – 14x + 91 = -6x + 27(distribute) -11x + 91 = -6x + 27(combine like terms) (subtract 91 from both sides) -11x = -6x – 64

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1-3: Solving Equations Solve 3x – 7(2x – 13) = 3(-2x + 9) 3x – 7(2x – 13) = 3(-2x + 9) 3x – 14x + 91 = -6x + 27(distribute) -11x + 91 = -6x + 27(combine like terms) (subtract 91 from both sides) -11x = -6x – 64 +6x +6x(add 6x to both sides) -5x = -64

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1-3: Solving Equations Solve 3x – 7(2x – 13) = 3(-2x + 9) 3x – 7(2x – 13) = 3(-2x + 9) 3x – 14x + 91 = -6x + 27(distribute) -11x + 91 = -6x + 27(combine like terms) (subtract 91 from both sides) -11x = -6x – 64 +6x +6x(add 6x to both sides) -5x = -64 -5 -5(divide both sides by -5) x = 12.8

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1-3: Solving Equations Solving a Formula for One of Its Variables The formula for the area of a trapezoid is A = ½ h(b 1 + b 2 ). Solve the formula for h. The goal is to use PEMDAS (in reverse) to get the variable in question alone. A = ½ h(b 1 + b 2 )

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1-3: Solving Equations Solving a Formula for One of Its Variables The formula for the area of a trapezoid is A = ½ h(b 1 + b 2 ). Solve the formula for h. The goal is to use PEMDAS (in reverse) to get the variable in question alone. A = ½ h(b 1 + b 2 ) x2 x2(multiply each side by 2, the reciprocal of ½) 2A = h(b 1 + b 2 )

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1-3: Solving Equations Solving a Formula for One of Its Variables The formula for the area of a trapezoid is A = ½ h(b 1 + b 2 ). Solve the formula for h. The goal is to use PEMDAS (in reverse) to get the variable in question alone. A = ½ h(b 1 + b 2 ) x2 x2(multiply each side by 2, the reciprocal of ½) 2A = h(b 1 + b 2 ) (b 1 + b 2 ) (b 1 + b 2 )(divide each side by b 1 + b 2 )

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1-3: Solving Equations Solving a Formula for One of Its Variables The formula for the area of a trapezoid is A = ½ h(b 1 + b 2 ). Solve the formula for b 1

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1-3: Solving Equations Solve for x

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1-3: Solving Equations Solve for x (multiply both sides by a, to clear the first denominator)

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1-3: Solving Equations Solve for x (multiply both sides by a, to clear the first denominator) (multiply both sides by b, to clear the second denominator)

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1-3: Solving Equations Solve for x (multiply both sides by a, to clear the first denominator) (multiply both sides by b, to clear the second denominator) (subtract bx on both sides, to get the x terms together)

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1-3: Solving Equations Solve for x (multiply both sides by a, to clear the first denominator) (multiply both sides by b, to clear the second denominator) (subtract bx on both sides, to get the x terms together) (distributive property, backwards)

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1-3: Solving Equations Solve for x (multiply both sides by a, to clear the first denominator) (multiply both sides by b, to clear the second denominator) (subtract bx on both sides, to get the x terms together) (distributive property, backwards) (divide both sides by “a – b”)

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1-3: Solving Equations Solve for x Are there any restrictions on the variables? Is there any number we couldn’t use in place of a, b, or x?

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1-3: Solving Equations Solve for x Are there any restrictions on the variables? Is there any number we couldn’t use in place of a, b, or x? Looking at the beginning problem a ≠ 0 and b ≠ 0 (can’t have a denominator of 0)

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1-3: Solving Equations Solve for x Are there any restrictions on the variables? Is there any number we couldn’t use in place of a, b, or x? Looking at the beginning problem a ≠ 0 and b ≠ 0 (can’t have a denominator of 0) Looking at the solution a – b ≠ 0(again, denominator can’t be 0) a ≠ b (add b to both sides)

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1-3: Solving Equations Assignment Page 21 1 – 27, odd problems Show your work Tomorrow: Word problems

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Chapter 1: Tools of Algebra 1-3: Solving Equations (Day 2) Essential Question: What is the procedure to solve an equation for a variable?

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1-3: Solving Equations Writing Equations to Solve Problems Ex 5: A dog kennel owner has 100 ft of fencing to enclose a rectangular dog run. She wants it to be 5 times as long as it is wide. Find the dimensions of the dog run. (Optional) Draw a diagram Determine the formula to use

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1-3: Solving Equations Writing Equations to Solve Problems Ex 5: A dog kennel owner has 100 ft of fencing to enclose a rectangular dog run. She wants it to be 5 times as long as it is wide. Find the dimensions of the dog run. (Optional) Draw a diagram Determine the formula to use Perimeter = 2 width + 2 length Determine the unknowns

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1-3: Solving Equations Writing Equations to Solve Problems Ex 5: A dog kennel owner has 100 ft of fencing to enclose a rectangular dog run. She wants it to be 5 times as long as it is wide. Find the dimensions of the dog run. (Optional) Draw a diagram Determine the formula to use Perimeter = 2 width + 2 length Determine the unknowns Let perimeter = 100 Let width = x Let length= 5x Use variable in the equation, and solve

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1-3: Solving Equations Perimeter = 2 width + 2 length 100 = 2 x + 2 5x

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1-3: Solving Equations Perimeter = 2 width + 2 length 100 = 2 x + 2 5x 100 = 2x + 10x

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1-3: Solving Equations Perimeter = 2 width + 2 length 100 = 2 x + 2 5x 100 = 2x + 10x 100 = 12x

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1-3: Solving Equations Perimeter = 2 width + 2 length 100 = 2 x + 2 5x 100 = 2x + 10x 100 = 12x 12 12 8 1 / 3 = x Determine both of your unknowns from the beginning of the problem

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1-3: Solving Equations Perimeter = 2 width + 2 length 100 = 2 x + 2 5x 100 = 2x + 10x 100 = 12x 12 12 8 1 / 3 = x Determine both of your unknowns from the beginning of the problem Width = x = 8 1 / 3 ft Length = 5x = / 3 = 41 2 / 3 ft

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1-3: Solving Equations Writing Equations to Solve Problems Ex 6: The lengths of the sides of a triangle are in the ratio 3:4:5. The perimeter of the triangle is 18 in. Find the length of the sides. (Optional) Draw a diagram Determine the formula to use

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1-3: Solving Equations Writing Equations to Solve Problems Ex 6: The lengths of the sides of a triangle are in the ratio 3:4:5. The perimeter of the triangle is 18 in. Find the length of the sides. (Optional) Draw a diagram Determine the formula to use Perimeter = s 1 + s 2 + s 3 Determine the variables

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1-3: Solving Equations Writing Equations to Solve Problems Ex 6: The lengths of the sides of a triangle are in the ratio 3:4:5. The perimeter of the triangle is 18 in. Find the length of the sides. (Optional) Draw a diagram Determine the formula to use Perimeter = s 1 + s 2 + s 3 Determine the variables Let perimeter = 18 Let s 1 (shortest side) = 3x Let s 2 (second side) = 4x Let s 3 (third side) = 5x Use variable in the equation, and solve

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1-3: Solving Equations Perimeter = s 1 + s 2 + s 3 18 = 3x + 4x + 5x

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1-3: Solving Equations Perimeter = s 1 + s 2 + s 3 18 = 3x + 4x + 5x 18 = 12x

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1-3: Solving Equations Perimeter = s 1 + s 2 + s 3 18 = 3x + 4x + 5x 18 = 12x 12 12 1.5 = x Determine both of your variables (unknowns) from the beginning of the problem

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1-3: Solving Equations Perimeter = s 1 + s 2 + s 3 18 = 3x + 4x + 5x 18 = 12x 12 12 1.5 = x Determine both of your variables (unknowns) from the beginning of the problem s 1 = 3x = = 4.5 in s 2 = 4x = = 6 in s 3 = 5x = = 7.5 in

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1-3: Solving Equations Assignment Page 22 29 – 35, all problems Skip 35b Show your work What equation you used to solve the problem Some of the steps you took to find your solution

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