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1 Topic 2.4.1 Applications of Linear Equations

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2 Topic 2.4.1 Applications of Linear Equations California Standards: 4.0: Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2 x – 5) + 4( x – 2) = 12. 5.0: Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. What it means for you: You’ll set up and solve equations that model real-life situations. Key words: application linear equation

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3 Topic 2.4.1 Applications of Linear Equations “Applications” are just “real-life” tasks. In this Topic, linear equations start to become really useful.

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4 Topic 2.4.1 Applications of Equations are “Real-Life” Tasks Applications of Linear Equations Applications questions are word problems that require you to set up and solve an equation. First decide how you will label the variables......then write the task out as an equation......making sure you include all the information given......then you can solve your equation. For example: Twice a number a plus 5 = 11. Find the value of a. To solve word problems, you need to take the following steps:

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5 Topic 2.4.1 Example 1 Solution follows… Applications of Linear Equations The sum of twice a number c and 7 is 21. Set up and solve an equation to find c. Solution You’re given the label “ c ” in the question — so just write out the equation. 2 c + 7 = 21 2 c = 14 c = 7

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6 Topic 2.4.1 Guided Practice Solution follows… Applications of Linear Equations 1. Twice a number c plus 17 is 31. Find the value of c. 2. Three times a number k minus 8 is 43. Find the number k. 3. The sum of four times a number m and 17 is the same as 7 less than six times the number m. Find the number m. 4. Seven minus five times the number x is equal to the sum of four times the number x and 25. Find the number x. c = 7 k = 17 m = 12 x = –2

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7 Topic 2.4.1 You Won’t Always Be Given the Variables Applications of Linear Equations Sometimes you’ll have to work out for yourself what the variables are, and decide on suitable labels for them.

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8 Topic 2.4.1 Example 2 Solution follows… Applications of Linear Equations Juanita’s age is 15 more than four times Vanessa’s age. The sum of their ages is 45. Set up and solve an equation to find their ages. Solution This time you have to decide for yourself how to label each term. Let v = Vanessa’s age 4 v + 15 = Juanita’s age v + (4 v + 15) = 45 The sum of their ages is 45 v + 4 v + 15 = 45 5 v + 15 = 45 5 v = 30 v = 6 Plug in the value for v to get Juanita’s age: 4 v + 15 = 4 × 6 + 15 = 24 + 15 = 39 So Vanessa is 6 years old and Juanita is 39 years old.

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9 Topic 2.4.1 Guided Practice Solution follows… Applications of Linear Equations 5. The length of a rectangular garden is 3 meters more than seven times its width. Find the length and width of the garden if the perimeter of the garden is 70 meters. 6. A rectangle is 4 meters longer than it is wide. The perimeter is 44 meters. What are the dimensions and area of the rectangle? Let w = width; w + 4 = length; 2( w + 4) + 2 w = perimeter So, 2( w + 4) + 2 w = 44 w = 9 Width = 9 m, Length = 9 + 4 = 13 m, Area = 9 m × 13 m = 117 m 2 Let w = width; 7 w + 3 = length; 2(7 w + 3) + 2 w = perimeter So, 2(7 w + 3) + 2 w = 70 16w = 64 w = 4 Width = 4 m, Length = 7 × 4 + 3 = 31 m

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10 Topic 2.4.1 Guided Practice Solution follows… Applications of Linear Equations 9. The sum of two consecutive integers is 117. What are the integers? Let n = 1 st integer; n + 1 = 2 nd integer So n + n + 1 = 117 — integers are 58 and 59 7. Abraham’s age is 4 less than half of Dominique’s age. Dominique’s age is 6 more than three times Juan’s age. The sum of their ages is 104. Find the age of each person. Let Abraham’s age be A, Dominique’s age be D, and Juan’s age be J. A = D – 4, D = 3 J + 6, J + D + A = 104 Writing in terms of J gives: J + (3 J + 6) + (3 J + 6) – 4 = 104 Abraham is 26, Dominique is 60, Juan is 18 1 2 1 2 8. The sides of an isosceles triangle are each 2 inches longer than the base. If the perimeter of the triangle is 97 inches, what are the lengths of the base and sides of the triangle? Let x = base length; ( x + 2) = side length; x + 2( x + 2) = perimeter So x + 2( x + 2) = 97 — base = 31 inches, sides = 33 inches each

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11 Topic 2.4.1 Independent Practice Solution follows… Applications of Linear Equations 1. Find the value of x, if the line segment shown on the right is 21 cm long. Also, find the length of each part of the line segment. 2. Point M is the midpoint of the line segment shown. Find the value of x and the length of the entire line segment. 3. The perimeter of the rectangular plot shown below is 142 feet. Find the dimensions of the plot. x = 17, segment is 182 units long x = 7, width = 30 feet, length = 41 feet x = 4, parts are 11 cm and 10 cm

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12 Topic 2.4.1 Independent Practice Solution follows… Applications of Linear Equations 5. The sum of one exterior angle at each vertex of any convex polygon is 360°. Find the size of each exterior angle shown around the triangle below. y = 35, angles are 115°, 115°, and 130° 4. The sum of the interior angles of a triangle is 180°. Find the size of each angle in the triangle shown below. x = 15, angles are 45°, 60°, and 75 °

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13 Topic 2.4.1 Independent Practice Solution follows… Applications of Linear Equations 6. The interior angles of a triangle sum to 180°. Find the size of each angle in the triangle on the right. 7. A rectangular garden has a length that is five meters less than three times its width. If the length is reduced by three meters and the width is reduced by one meter, the perimeter will be 62 meters. Find the dimensions of the garden. x = 25, angles are 25°, 75°, and 80° Dimensions are 10 m × 25 m

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14 Topic 2.4.1 Applications of Linear Equations Round Up The thing to do with any word problem is to write out a math equation that describes the same situation. Then you can use all the techniques you’ve learned to solve the equation.

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