Create Equations

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Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 2

1. Kaia is designing a small vegetable garden. She wants to create a rectangular plot where the length is 9 meters more than twice the width. The total perimeter of the garden is to be 84 meters. Write an equation in two variables to represent the relationship between the length L and the width W, and then determine the dimensions of the garden. Answer: Length = 31 meters, Width = 11 meters Solution: Let L represent the length and W represent the width. The perimeter P of a rectangle is given by P = 2L + 2W. We are told the perimeter is 84 meters, so the first equation is: 2L + 2W = 84.
   Full step-by-step solution

   Step 1: Let L represent the length and W represent the width. The perimeter P of a rectangle is given by P = 2L + 2W. We are told the perimeter is 84 meters, so the first equation is: 2L + 2W = 84. Step 2: The length is 9 meters more than twice the width, so the second equation is: L = 2W + 9. Step 3: Substitute L = 2W + 9 into the perimeter equation: 2(2W + 9) + 2W = 84. Step 4: Simplify: 4W + 18 + 2W = 84 → 6W + 18 = 84. Step 5: Subtract 18 from both sides: 6W = 66. Step 6: Divide both sides by 6: W = 11. Step 7: Now find L using L = 2W + 9: L = 2(11) + 9 = 22 + 9 = 31. Step 8: The dimensions are: width = 11 meters, length = 31 meters. The answer is Length = 31 meters, Width = 11 meters.

2. Kaia is organizing a fundraising concert. She rents a venue that costs a fixed fee of $450. Additionally, she needs to pay for security guards, where each guard costs $85, and for sound equipment, which costs a flat rate of $220 plus $15 per hour of use. Write an equation in two or more variables that represents the total cost C of the event in terms of the number of security guards g and the number of hours h the sound equipment is used. Answer: C = 85g + 15h + 670 Solution: Identify all fixed costs: venue fee $450, and sound equipment flat rate $220. Total fixed cost = 450 + 220 = 670. Variable cost for security guards: each guard costs $85, so for g guards, cost = 85g.
   Full step-by-step solution

   Step 1: Identify all fixed costs: venue fee $450, and sound equipment flat rate $220. Total fixed cost = 450 + 220 = 670. Step 2: Variable cost for security guards: each guard costs $85, so for g guards, cost = 85g. Step 3: Variable cost for sound equipment: $15 per hour, so for h hours, cost = 15h. Step 4: Total cost C is the sum of all fixed and variable costs: C = 85g + 15h + 670. Final answer: C = 85g + 15h + 670.

3. Olivia is planning a school fundraiser where she will sell two types of tickets: adult tickets at $15 each and student tickets at $10 each. She needs to raise a total of $2,500. Let x represent the number of adult tickets sold and y represent the number of student tickets sold. Write an equation in two variables that models the total revenue from ticket sales. If she sells 120 adult tickets, how many student tickets must she sell to reach her goal? Answer: 70 Solution: Revenue from adult tickets is 15 dollars per ticket times x tickets, so 15x. Revenue from student tickets is 10 dollars per ticket times y tickets, so 10y.
   Full step-by-step solution

   Step 1: Revenue from adult tickets is 15 dollars per ticket times x tickets, so 15x. Step 2: Revenue from student tickets is 10 dollars per ticket times y tickets, so 10y. Step 3: Total revenue must be $2,500, so the equation is 15x + 10y = 2500. Step 4: Substitute x = 120 into the equation: 15(120) + 10y = 2500. Step 5: Simplify: 1800 + 10y = 2500. Step 6: Subtract 1800 from both sides: 10y = 700. Step 7: Divide both sides by 10: y = 70. Step 8: Olivia must sell 70 student tickets to reach her goal.

4. Mere is designing a rectangular mural for her school's entrance hall. The mural's length is 4 meters more than twice its width. The total area of the mural is 126 square meters. Write an equation in two variables that represents the relationship between the length (l) and width (w) of the mural, and then express the total area constraint as a second equation. Finally, use both equations to determine the dimensions of the mural. Answer: width = 7 meters, length = 18 meters Solution: Let w represent the width (in meters) and l represent the length (in meters). Translate 'length is 4 meters more than twice its width' into an equation: l = 2w + 4. The area of a rectangle is length times width.
   Full step-by-step solution

   Step 1: Let w represent the width (in meters) and l represent the length (in meters). Step 2: Translate 'length is 4 meters more than twice its width' into an equation: l = 2w + 4. Step 3: The area of a rectangle is length times width. The area is 126 square meters. So, the second equation is: l * w = 126. Step 4: Substitute the expression for l from the first equation into the second equation: (2w + 4) * w = 126. Step 5: Expand and simplify: 2w^2 + 4w = 126. Step 6: Rearrange into a standard quadratic equation: 2w^2 + 4w - 126 = 0. Step 7: Divide the entire equation by 2 to simplify: w^2 + 2w - 63 = 0. Step 8: Factor the quadratic: (w + 9)(w - 7) = 0. Step 9: Solve for w: w + 9 = 0 gives w = -9, and w - 7 = 0 gives w = 7. Step 10: Width cannot be negative, so w = 7 meters. Step 11: Substitute w = 7 into the first equation to find the length: l = 2(7) + 4 = 14 + 4 = 18 meters. The answer is width = 7 meters, length = 18 meters.

5. Ava is designing a triangular garden where the area is 84 square meters. The base is 7 meters more than the height. Write an equation relating the area to the base and height. Answer: A = (1/2)bh where b = h + 7 Solution: The area of a triangle is calculated as one-half times base times height. When one dimension is expressed in terms of the other, we can create an equation with two variables that represents their relationship.
6. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length of the garden to be 4 meters more than twice its width. Write a system of equations to represent this situation and solve for the dimensions of the garden. Answer: width = 6 meters, length = 14 meters Solution: When solving geometry problems with unknown dimensions, you can create equations using known formulas like perimeter.
   Full step-by-step solution

   When solving geometry problems with unknown dimensions, you can create equations using known formulas like perimeter. If one dimension depends on another, you can substitute that relationship into your equation to solve for both variables systematically.