Create Equations

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Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 2

1. Emma is mixing two types of fertilizer for her garden. She needs a total of 21 kilograms of the mixture. Type A costs $7 per kilogram, and Type B costs $11 per kilogram. She has a budget of $187 for the fertilizer. Write a system of equations to represent this situation, then determine how many kilograms of each type she should buy. Answer: 11 kilograms of Type A and 10 kilograms of Type B Solution: Define the variables. Let x = number of kilograms of Type A fertilizer. Let y = number of kilograms of Type B fertilizer.
   Full step-by-step solution

   Step 1: Define the variables. Let x = number of kilograms of Type A fertilizer. Let y = number of kilograms of Type B fertilizer. Step 2: Write the equation for total weight: x + y = 21. Step 3: Write the equation for total cost: 7x + 11y = 187. Step 4: Solve the system using substitution or elimination. Use substitution: from x + y = 21, we get y = 21 - x. Step 5: Substitute into the cost equation: 7x + 11(21 - x) = 187. Step 6: Simplify: 7x + 231 - 11x = 187. Step 7: Combine like terms: -4x + 231 = 187. Step 8: Subtract 231 from both sides: -4x = -44. Step 9: Divide by -4: x = 11. Step 10: Substitute x = 11 into y = 21 - x: y = 21 - 11 = 10. Step 11: Check: 11 + 10 = 21 kg, and 7(11) + 11(10) = 77 + 110 = 187 dollars. The answer is 11 kilograms of Type A and 10 kilograms of Type B.

2. 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.

3. Matiu is designing a rectangular garden with a path around it. The garden itself has length L meters and width W meters. The path surrounding the garden has a uniform width of 2 meters on all sides. Write an equation for the total area A of the garden plus the path, in terms of L and W. Answer: A = (L + 4)(W + 4) Solution: The garden is a rectangle with length L and width W. A path of uniform width 2 meters surrounds the garden on all sides.
   Full step-by-step solution

   Step 1: The garden is a rectangle with length L and width W. Step 2: A path of uniform width 2 meters surrounds the garden on all sides. This means the path adds 2 meters to the left, 2 meters to the right, 2 meters to the top, and 2 meters to the bottom. Step 3: Therefore, the total length of the garden plus path is L + 2 + 2 = L + 4. Step 4: The total width of the garden plus path is W + 2 + 2 = W + 4. Step 5: The total area A (garden + path) is the area of the outer rectangle: total length times total width. Step 6: So the equation is A = (L + 4)(W + 4). The answer is A = (L + 4)(W + 4).

4. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length to be 4 meters more than twice the 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, we often use variables to represent the measurements we're trying to find.
   Full step-by-step solution

   When solving geometry problems with unknown dimensions, we often use variables to represent the measurements we're trying to find. The perimeter formula for rectangles provides one equation, while the relationship between length and width gives us another. By substituting one equation into the other, we can solve for the unknown values systematically.

5. Isabella is designing a triangular garden with a base that is 8 meters longer than twice its height. Write an equation for the area A of the garden in terms of the height h. Answer: A = h(2h + 8)/2 Solution: The area of a triangle is calculated as one-half times base times height.
   Full step-by-step solution

   The area of a triangle is calculated as one-half times base times height. When the base is expressed in terms of height using a given relationship, you can substitute that expression into the area formula to create an equation with two variables.

6. A scientist is studying bacterial growth in a lab. The population P(t) of bacteria after t hours is modeled by the exponential function P(t) = 500 × 2^(t/3). Meanwhile, the amount of nutrient solution N(t) in milliliters decreases according to the linear function N(t) = 200 - 15t. At what time will the bacteria population reach exactly 4000, and how much nutrient solution will remain at that time? Answer: 9 hours, 65 milliliters Solution: Set up the equation for when bacteria population reaches 4000: 500 × 2^(t/3) = 4000 Divide both sides by 500: 2^(t/3) = 8 Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Set exponents equal: t/3 = 3 Multiply both sides by 3: t = 9 hours Substitute t = 9 into the nutrient function: N(9) = 200 - 15 × 9…
   Full step-by-step solution

   Step 1: Set up the equation for when bacteria population reaches 4000: 500 × 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Step 4: Set exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 hours Step 6: Substitute t = 9 into the nutrient function: N(9) = 200 - 15 × 9 Step 7: Calculate: 200 - 135 = 65 milliliters Step 8: The bacteria reach 4000 after 9 hours with 65 milliliters of nutrient solution remaining.

7. Olivia is designing a rectangular community garden. The length of the garden is 7 meters more than three times its width. If the perimeter of the garden is 94 meters, write a system of two equations in two variables to represent this situation, and then find the length and width of the garden. Answer: width = 10 meters, length = 37 meters Solution: Let w represent the width in meters and l represent the length in meters. Step 2: From the problem, the length is 7 more than three times the width, so l = 3w + 7. Step 3: The perimeter of a rectangle is P = 2l + 2w.
   Full step-by-step solution

   Step 1: Let w represent the width in meters and l represent the length in meters. Step 2: From the problem, the length is 7 more than three times the width, so l = 3w + 7. Step 3: The perimeter of a rectangle is P = 2l + 2w. We are given P = 94, so 2l + 2w = 94. Step 4: Substitute l = 3w + 7 into the perimeter equation: 2(3w + 7) + 2w = 94. Step 5: Simplify: 6w + 14 + 2w = 94, then 8w + 14 = 94. Step 6: Subtract 14 from both sides: 8w = 80. Step 7: Divide by 8: w = 10. Step 8: Substitute w = 10 into l = 3w + 7: l = 3(10) + 7 = 30 + 7 = 37. Step 9: The width is 10 meters and the length is 37 meters. The answer is width = 10 meters, length = 37 meters.

8. Emma is designing a rectangular garden where the length is 10 meters more than twice the width. The perimeter of the garden is 80 meters. Write an equation in two variables that relates the length (l) and width (w) to the perimeter. Answer: 2l + 2w = 80; l = 2w + 10 Solution: The perimeter of a rectangle is given by P = 2l + 2w. Here, P = 80, so the first equation is 2l + 2w = 80. The problem states the length is 10 meters more than twice the width.
   Full step-by-step solution

   Step 1: The perimeter of a rectangle is given by P = 2l + 2w. Here, P = 80, so the first equation is 2l + 2w = 80. Step 2: The problem states the length is 10 meters more than twice the width. This gives the second equation: l = 2w + 10. Step 3: These two equations together form a system that can be solved to find l and w. The answer is the system: 2l + 2w = 80 and l = 2w + 10.