Create Equations

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Create Equations · Grade 9 · Worksheet 1

1. Emma is building a rectangular garden in her backyard. The garden is divided into two sections: one for vegetables and one for flowers. The vegetable section is a square with side length x meters. The flower section is a rectangle with width y meters and length that is 5 meters longer than the vegetable section's side length. The total area of the garden is 70 square meters. Write an equation in two variables that represents the total area of the garden. Answer: x^2 + x(y + 5) = 70 Solution: The vegetable section is a square with side length x. Its area is x * x = x^2. The flower section is a rectangle with width y and length that is 5 meters longer than the vegetable section's side length.
   Full step-by-step solution

   Step 1: The vegetable section is a square with side length x. Its area is x * x = x^2. Step 2: The flower section is a rectangle with width y and length that is 5 meters longer than the vegetable section's side length. So its length is x + 5. Step 3: The area of the flower section is width times length: y * (x + 5). Step 4: The total area of the garden is the sum of the areas of both sections: x^2 + y(x + 5). Step 5: The problem states the total area is 70 square meters, so the equation is x^2 + y(x + 5) = 70. The answer is x^2 + y(x + 5) = 70.

2. Aisha is analyzing the growth of bacteria in a lab experiment. The bacteria population follows the exponential model P(t) = 500 * e^(0.03t), where t is time in hours. She needs to determine when the population will reach 1000 bacteria. Which equation should she solve to find this time? Answer: A. 500 * e^(0.03t) = 1000 Solution: The population function is given as P(t) = 500 * e^(0.03t) We want to find when P(t) = 1000 Substitute 1000 for P(t): 500 * e^(0.03t) = 1000 This is the equation Aisha needs to solve to find the time when the population reaches 1000 bacteria The correct equation is 500 * e^(0.03t) = 1000, which…
   Full step-by-step solution

   Step 1: The population function is given as P(t) = 500 * e^(0.03t) Step 2: We want to find when P(t) = 1000 Step 3: Substitute 1000 for P(t): 500 * e^(0.03t) = 1000 Step 4: This is the equation Aisha needs to solve to find the time when the population reaches 1000 bacteria The correct equation is 500 * e^(0.03t) = 1000, which corresponds to choice C.

3. The sum of two numbers is 24. When the larger number is squared and then decreased by the square of the smaller number, the result is 96. What is the larger number? Answer: 14 Solution: Write the equations from the problem. From "The sum of two numbers is 24": L + S = 24 From "When the larger number is squared and then decreased by the square of the smaller number, the result is 96": L^2 - S^2 = 96 Notice that L^2 - S^2 is a difference of squares.
   Full step-by-step solution

   Let's call the larger number L and the smaller number S. Step 1: Write the equations from the problem. From "The sum of two numbers is 24": L + S = 24 From "When the larger number is squared and then decreased by the square of the smaller number, the result is 96": L^2 - S^2 = 96 Step 2: Notice that L^2 - S^2 is a difference of squares. We can factor it: L^2 - S^2 = (L - S)(L + S) = 96 Step 3: Substitute L + S = 24 into the factored equation. (L - S)(24) = 96 Step 4: Solve for (L - S). Divide both sides by 24: L - S = 96 / 24 L - S = 4 Step 5: Now we have a system of two equations: Equation 1: L + S = 24 Equation 2: L - S = 4 Step 6: Add the two equations to eliminate S. (L + S) + (L - S) = 24 + 4 L + S + L - S = 28 2L = 28 Step 7: Solve for L. L = 28 / 2 L = 14 Step 8: Find S for completeness (optional check). From L + S = 24, 14 + S = 24, so S = 10. Step 9: Check the conditions. Sum: 14 + 10 = 24 (correct) L^2 - S^2 = 196 - 100 = 96 (correct) Therefore, the larger number is 14.

4. Noah is drawing a rectangular prism on a coordinate grid. The base of the prism is a rectangle with vertices at (1, 6), (6, 6), (6, 1), and (1, 1). The height of the prism is represented by the variable h. Write an equation in two variables for the total surface area, S, of the rectangular prism. Answer: S = 2lw + 2lh + 2wh or S = 2(5)(5) + 2(5)h + 2(5)h = 50 + 20h Solution: Find the dimensions of the base rectangle. The vertices are (1,6), (6,6), (6,1), and (1,1). The length (horizontal distance) is from x=1 to x=6, which is 5 units.
   Full step-by-step solution

   Step 1: Find the dimensions of the base rectangle. The vertices are (1,6), (6,6), (6,1), and (1,1). The length (horizontal distance) is from x=1 to x=6, which is 5 units. The width (vertical distance) is from y=1 to y=6, which is 5 units. So, the base is a 5 by 5 square. Let l = 5 and w = 5. Step 2: The height of the prism is h (the variable). Step 3: The surface area S of a rectangular prism is given by the formula: S = 2lw + 2lh + 2wh. Step 4: Substitute l = 5 and w = 5 into the formula: S = 2(5)(5) + 2(5)h + 2(5)h. Step 5: Simplify the equation: S = 2(25) + 10h + 10h = 50 + 20h. The final equation is S = 50 + 20h.

5. Liam is planning a rectangular garden. The length of the garden is 5 meters more than twice its width. He wants to put a fence around the entire garden, which will cost $15 per meter. Write an equation for the total cost, C, in terms of the width, w, of the garden. Answer: C = 30(2w + 5) + 30w or C = 90w + 150 Solution: Let w represent the width of the garden in meters. The length is 5 more than twice the width, so length = 2w + 5. The perimeter P of a rectangle is 2 * length + 2 * width.
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in meters. Step 2: The length is 5 more than twice the width, so length = 2w + 5. Step 3: The perimeter P of a rectangle is 2 * length + 2 * width. P = 2(2w + 5) + 2w P = 4w + 10 + 2w P = 6w + 10 meters. Step 4: The cost per meter is $15, so total cost C = 15 * P. C = 15 * (6w + 10) C = 90w + 150. The equation for the total cost in terms of width is C = 90w + 150.

6. Noah is designing a rectangular solar panel array for a rooftop. The length of the array is 7 meters more than twice its width. The total perimeter of the array is 62 meters. Write an equation in two variables to represent this situation, and determine the dimensions (length and width) of the solar panel array. Answer: Width = 8 meters, Length = 23 meters Solution: Let w represent the width and l represent the length. From the problem, length is 7 meters more than twice the width: l = 2w + 7. The perimeter P of a rectangle is P = 2l + 2w.
   Full step-by-step solution

   Step 1: Let w represent the width and l represent the length. From the problem, length is 7 meters more than twice the width: l = 2w + 7. Step 2: The perimeter P of a rectangle is P = 2l + 2w. We are given P = 62, so: 2l + 2w = 62. Step 3: Substitute l = 2w + 7 into the perimeter equation: 2(2w + 7) + 2w = 62. Step 4: Distribute: 4w + 14 + 2w = 62. Step 5: Combine like terms: 6w + 14 = 62. Step 6: Subtract 14 from both sides: 6w = 48. Step 7: Divide by 6: w = 8. Step 8: Substitute w = 8 into l = 2w + 7: l = 2(8) + 7 = 16 + 7 = 23. Step 9: The width is 8 meters and the length is 23 meters. The answer is Width = 8 meters, Length = 23 meters.

7. Isabella is designing a rectangular garden with a perimeter of 84 meters. The length is 8 meters more than the width. Write an equation in terms of length (l) and width (w) that represents the perimeter. Answer: 2l + 2w = 84 Solution: The perimeter of a rectangle is calculated by adding all four sides. Since opposite sides are equal, the perimeter is twice the sum of length and width.
   Full step-by-step solution

   The perimeter of a rectangle is calculated by adding all four sides. Since opposite sides are equal, the perimeter is twice the sum of length and width. When one dimension is expressed in terms of the other, it creates a relationship that can be used to form an equation with two variables.