Linear Systems

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Linear Systems · Grade 12 · Worksheet 3

1. A chemical engineering company is optimizing a reactor that uses three catalysts (X, Y, Z) in different trials. In Trial 1, using 3 units of X, 2 units of Y, and 1 unit of Z produced 34 grams of product. In Trial 2, using 1 unit of X, 4 units of Y, and 2 units of Z produced 36 grams. In Trial 3, using 2 units of X, 1 unit of Y, and 3 units of Z produced 35 grams. The effectiveness of each catalyst can be represented by coefficients a, b, and c respectively in the equation: (catalyst amount) × (catalyst effectiveness) = product yield. Determine the effectiveness coefficient for each catalyst. Answer: a = 8, b = 5, c = 3 Solution: In systems of equations with three variables, each condition provides one equation. The coefficients in each equation come from the amounts used in each trial, and the constants are the product yields.
   Full step-by-step solution

   In systems of equations with three variables, each condition provides one equation. The coefficients in each equation come from the amounts used in each trial, and the constants are the product yields. Solving such systems often involves elimination or substitution methods to find the values that satisfy all equations simultaneously.

2. Aroha, Tane, and Kaia are three friends who run a small business selling handmade candles in three different scents: lavender, vanilla, and cinnamon. They sell their candles in gift boxes. For one gift box, the number of lavender candles is three times the number of vanilla candles, and the number of cinnamon candles is 7 more than the number of vanilla candles. The total number of candles in the box is 47. Determine how many of each scent are in the gift box. Answer: lavender = 24, vanilla = 8, cinnamon = 15 Solution: Let x be the number of vanilla candles. Then lavender = 3x, and cinnamon = x + 7. The total is x + 3x + (x + 7) = 47.
   Full step-by-step solution

   Let x be the number of vanilla candles. Then lavender = 3x, and cinnamon = x + 7. The total is x + 3x + (x + 7) = 47. Combine like terms: 5x + 7 = 47. Subtract 7 from both sides: 5x = 40. Divide by 5: x = 8. So vanilla = 8, lavender = 3(8) = 24, cinnamon = 8 + 7 = 15. The answer is lavender = 24, vanilla = 8, cinnamon = 15.

3. A chemical engineering company is designing a reactor system where three substances (A, B, and C) must be mixed in specific ratios. Substance A contains 2 units of compound X and 1 unit of compound Y per liter. Substance B contains 1 unit of compound X, 3 units of compound Y, and 2 units of compound Z per liter. Substance C contains 4 units of compound X, 1 unit of compound Y, and 1 unit of compound Z per liter. To achieve the optimal reaction, the final mixture requires exactly 14 units of compound X, 9 units of compound Y, and 5 units of compound Z. How many liters of each substance (A, B, and C) should be used? Answer: A = 2, B = 1, C = 2 Solution: When solving mixture problems with multiple components, we create equations that account for how much each substance contributes to the total amount of each compound.
   Full step-by-step solution

   When solving mixture problems with multiple components, we create equations that account for how much each substance contributes to the total amount of each compound. The coefficients in these equations come from the composition of each substance, and the constants represent the desired total amounts. Solving such systems helps determine the precise quantities needed to achieve specific chemical or material properties.

4. Solve the system: 3x + 2y - z = 4, 2x - y + 3z = 7, x + 4y - 2z = -1 Answer: (1, -1, 2) Solution: 3x + 2y - z = 4 (1) 2x - y + 3z = 7 (2) x + 4y - 2z = -1 (3) Eliminate z from equations (1) and (2): Multiply (1) by 3: 9x + 6y - 3z = 12 Add to (2): (9x + 6y - 3z) + (2x - y + 3z) = 12 + 7 11x + 5y = 19 (4) Eliminate z from equations (1) and (3): Multiply (1) by 2: 6x + 4y - 2z = 8 Subtract…
   Full step-by-step solution

   Step 1: Write the system: 3x + 2y - z = 4 (1) 2x - y + 3z = 7 (2) x + 4y - 2z = -1 (3) Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 3: 9x + 6y - 3z = 12 Add to (2): (9x + 6y - 3z) + (2x - y + 3z) = 12 + 7 11x + 5y = 19 (4) Step 3: Eliminate z from equations (1) and (3): Multiply (1) by 2: 6x + 4y - 2z = 8 Subtract (3): (6x + 4y - 2z) - (x + 4y - 2z) = 8 - (-1) 5x = 9 x = 9/5 Step 4: Substitute x = 9/5 into equation (4): 11(9/5) + 5y = 19 99/5 + 5y = 19 5y = 19 - 99/5 5y = 95/5 - 99/5 5y = -4/5 y = -4/25 Step 5: Substitute x = 9/5 and y = -4/25 into equation (1): 3(9/5) + 2(-4/25) - z = 4 27/5 - 8/25 - z = 4 135/25 - 8/25 - z = 100/25 127/25 - z = 100/25 -z = 100/25 - 127/25 -z = -27/25 z = 27/25 Step 6: Verify the solution: Equation (1): 3(9/5) + 2(-4/25) - (27/25) = 27/5 - 8/25 - 27/25 = 135/25 - 8/25 - 27/25 = 100/25 = 4 ✓ Equation (2): 2(9/5) - (-4/25) + 3(27/25) = 18/5 + 4/25 + 81/25 = 90/25 + 4/25 + 81/25 = 175/25 = 7 ✓ Equation (3): 9/5 + 4(-4/25) - 2(27/25) = 9/5 - 16/25 - 54/25 = 45/25 - 16/25 - 54/25 = -25/25 = -1 ✓ The solution is (9/5, -4/25, 27/25).

5. Aroha, Tane, and Kaia are managing a community recycling program that collects three types of materials: paper, plastic, and glass. Last month, the total weight of all collected materials was 216 kilograms. The weight of paper collected was 15 kilograms more than twice the weight of plastic collected. The weight of glass collected was 18 kilograms less than the sum of the weights of paper and plastic collected. Determine the weight (in kilograms) of each type of material collected. Answer: paper = 87 kg, plastic = 36 kg, glass = 93 kg Solution: Define variables. Let p = weight of paper (kg), r = weight of plastic (kg), g = weight of glass (kg).
   Full step-by-step solution

   Step 1: Define variables. Let p = weight of paper (kg), r = weight of plastic (kg), g = weight of glass (kg). Step 2: Write the equations from the problem: Equation 1 (total): p + r + g = 216 Equation 2 (paper vs plastic): p = 2r + 15 Equation 3 (glass vs paper and plastic): g = p + r - 18 Step 3: Substitute equation 2 into equation 3: g = (2r + 15) + r - 18 = 3r - 3 Step 4: Substitute p = 2r + 15 and g = 3r - 3 into equation 1: (2r + 15) + r + (3r - 3) = 216 Step 5: Combine like terms: 6r + 12 = 216 Step 6: Subtract 12 from both sides: 6r = 204 Step 7: Divide by 6: r = 34 Step 8: Find p: p = 2(34) + 15 = 68 + 15 = 83 Step 9: Find g: g = 3(34) - 3 = 102 - 3 = 99 Step 10: Check total: 83 + 34 + 99 = 216. The conditions hold. Final answer: paper = 83 kg, plastic = 34 kg, glass = 99 kg.