Linear Systems

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Linear Systems · Grade 12 · Worksheet 1

1. A chemistry lab is conducting three experiments to determine the heat absorption rates (in J/g) of three unknown compounds: X, Y, and Z. In Experiment 1, a sample containing 1 gram of X, 3 grams of Y, and 1 gram of Z absorbs 31 J of heat. In Experiment 2, a sample containing 2 grams of X, 1 gram of Y, and 6 grams of Z absorbs 46 J of heat. In Experiment 3, a sample containing 1 gram of X, 1 gram of Y, and 1 gram of Z absorbs 16 J of heat. If the total heat absorbed is the sum of the heat absorbed by each compound individually (mass × heat absorption rate), determine the heat absorption rate (in J/g) for each compound. Answer: x = 6, y = 7, z = 3 Solution: Set up the system of equations. Experiment 1: 1x + 3y + 1z = 31 Experiment 2: 2x + 1y + 6z = 46 Experiment 3: 1x + 1y + 1z = 16 Use elimination.
   Full step-by-step solution

   Step 1: Set up the system of equations. Experiment 1: 1x + 3y + 1z = 31 Experiment 2: 2x + 1y + 6z = 46 Experiment 3: 1x + 1y + 1z = 16 Step 2: Use elimination. Subtract the third equation from the first equation to eliminate x and z: (1x + 3y + 1z) - (1x + 1y + 1z) = 31 - 16 2y = 15 y = 7.5 Step 3: Subtract the third equation from the second equation to eliminate x and z: (2x + 1y + 6z) - (1x + 1y + 1z) = 46 - 16 1x + 5z = 30 Step 4: Substitute y = 7.5 into the third equation: 1x + 7.5 + 1z = 16 x + z = 8.5 Step 5: Now we have two equations: x + 5z = 30 x + z = 8.5 Subtract the second from the first: (x + 5z) - (x + z) = 30 - 8.5 4z = 21.5 z = 5.375 Step 6: Substitute z = 5.375 into x + z = 8.5: x + 5.375 = 8.5 x = 3.125 Step 7: Verify with the first equation: 1(3.125) + 3(7.5) + 1(5.375) = 3.125 + 22.5 + 5.375 = 31 ✓ The heat absorption rates are: x = 3.125 J/g, y = 7.5 J/g, z = 5.375 J/g.

2. A chemical engineering company is designing a reactor system where three substances (A, B, and C) must be mixed in precise ratios. The concentration equations are: 2x + 3y - z = 15 for substance A, x - 2y + 4z = 10 for substance B, and 3x + y - 2z = 8 for substance C. What are the exact concentrations (x, y, z) that satisfy all three equations simultaneously? Answer: (4, 3, 2) Solution: Solving such systems involves finding the point where all three planes intersect in three-dimensional space.
   Full step-by-step solution

   Systems of three linear equations can model many real-world scenarios like chemical mixtures, economic models, or engineering designs. Solving such systems involves finding the point where all three planes intersect in three-dimensional space. Common methods include elimination (adding or subtracting equations to eliminate variables) or substitution (solving one equation for a variable and substituting into others). These methods systematically reduce the complexity until all variable values are determined.

3. Solve the system: 2x + y - z = 8, x - 3y + 2z = -1, 3x + 2y - 4z = 9 Answer: x = 3, y = 2, z = 0 Solution: Systems of three equations with three variables can be solved using elimination or substitution methods.
   Full step-by-step solution

   Systems of three equations with three variables can be solved using elimination or substitution methods. The goal is to systematically reduce the system to two equations with two variables, then to one equation with one variable. This approach works because each equation represents a plane in three-dimensional space, and the solution is the point where all three planes intersect.

4. A civil engineering firm is designing a suspension bridge where the main cable follows a parabolic path. The cable's height above the roadway at three different horizontal positions is measured: at x = 0 meters, the height is 40 meters; at x = 20 meters, the height is 30 meters; and at x = 40 meters, the height is 60 meters. The cable's shape can be modeled by the quadratic equation y = ax² + bx + c, where y is the height in meters and x is the horizontal distance in meters. Determine the coefficients a, b, and c that define the cable's precise shape. Answer: a = 0.1, b = -3, c = 40 Solution: Substitute the first point (0,40) into y = ax² + bx + c: 40 = a(0)² + b(0) + c 40 = c Substitute the second point (20,30) into y = ax² + bx + c: 30 = a(20)² + b(20) + 40 30 = 400a + 20b + 40 400a + 20b = -10 Substitute the third point (40,60) into y = ax² + bx + c: 60 = a(40)² + b(40) + 40 60 =…
   Full step-by-step solution

   Step 1: Substitute the first point (0,40) into y = ax² + bx + c: 40 = a(0)² + b(0) + c 40 = c Step 2: Substitute the second point (20,30) into y = ax² + bx + c: 30 = a(20)² + b(20) + 40 30 = 400a + 20b + 40 400a + 20b = -10 Step 3: Substitute the third point (40,60) into y = ax² + bx + c: 60 = a(40)² + b(40) + 40 60 = 1600a + 40b + 40 1600a + 40b = 20 Step 4: Solve the system of equations: From Step 2: 400a + 20b = -10 From Step 3: 1600a + 40b = 20 Multiply the first equation by 2: 800a + 40b = -20 Subtract this from the second equation: (1600a + 40b) - (800a + 40b) = 20 - (-20) 800a = 40 a = 40/800 = 0.05 Step 5: Substitute a = 0.05 into 400a + 20b = -10: 400(0.05) + 20b = -10 20 + 20b = -10 20b = -30 b = -1.5 Step 6: We already found c = 40 from Step 1 Step 7: Verify with the third point: y = 0.05(40)² + (-1.5)(40) + 40 y = 0.05(1600) - 60 + 40 y = 80 - 60 + 40 = 60 ✓ The coefficients are: a = 0.05, b = -1.5, c = 40

5. A chemical engineering company is testing three different catalysts (A, B, and C) for a reaction. In Trial 1, using 2 units of A, 1 unit of B, and 3 units of C produced 23 grams of product. In Trial 2, using 4 units of A, 2 units of B, and 2 units of C produced 32 grams. In Trial 3, using 1 unit of A, 3 units of B, and 1 unit of C produced 15 grams. The effectiveness of each catalyst can be represented by coefficients x, y, and z respectively in the equation: (catalyst amount) × (catalyst effectiveness) = product yield. Determine the effectiveness coefficient for each catalyst. Answer: A: 4, B: 2, C: 5 Solution: In many scientific and engineering applications, we encounter situations where multiple factors contribute to an outcome.
   Full step-by-step solution

   In many scientific and engineering applications, we encounter situations where multiple factors contribute to an outcome. By conducting different experiments with varying combinations, we can mathematically determine the individual contribution of each factor. This approach is fundamental in fields like chemistry, economics, and data analysis where we need to isolate the effects of different variables in complex systems.

6. 2x + 3y - z = 11, x - 2y + 3z = 6, 3x + y - 2z = 5 Answer: x = 3, y = 2, z = 1 Solution: Multiply the second equation by 2: 2(x - 2y + 3z) = 2(6) → 2x - 4y + 6z = 12 Subtract the first equation from this result: (2x - 4y + 6z) - (2x + 3y - z) = 12 - 11 → -7y + 7z = 1 Multiply the second equation by 3: 3(x - 2y + 3z) = 3(6) → 3x - 6y + 9z = 18 Subtract the third equation from this…
   Full step-by-step solution

   Step 1: Multiply the second equation by 2: 2(x - 2y + 3z) = 2(6) → 2x - 4y + 6z = 12 Step 2: Subtract the first equation from this result: (2x - 4y + 6z) - (2x + 3y - z) = 12 - 11 → -7y + 7z = 1 Step 3: Multiply the second equation by 3: 3(x - 2y + 3z) = 3(6) → 3x - 6y + 9z = 18 Step 4: Subtract the third equation from this result: (3x - 6y + 9z) - (3x + y - 2z) = 18 - 5 → -7y + 11z = 13 Step 5: Now we have: -7y + 7z = 1 and -7y + 11z = 13 Step 6: Subtract the first from the second: (-7y + 11z) - (-7y + 7z) = 13 - 1 → 4z = 12 → z = 3 Step 7: Substitute z = 3 into -7y + 7z = 1: -7y + 7(3) = 1 → -7y + 21 = 1 → -7y = -20 → y = 20/7 Step 8: Substitute y = 20/7 and z = 3 into the first equation: 2x + 3(20/7) - 3 = 11 → 2x + 60/7 - 3 = 11 → 2x + 60/7 - 21/7 = 11 → 2x + 39/7 = 11 → 2x = 11 - 39/7 → 2x = 77/7 - 39/7 → 2x = 38/7 → x = 19/7 Final answer: x = 19/7, y = 20/7, z = 3