Linear Systems

lessonbunny.com

Answer Key & Explanations

Linear Systems · Grade 12 · Worksheet 2

1. 3x + 2y - z = 11, 2x - y + 3z = 7, x + 4y - 2z = 5 Answer: x = 3, y = 1, z = 2 Solution: Step 1: Label the equations: (1) 3x + 2y - z = 11 (2) 2x - y + 3z = 7 (3) x + 4y - 2z = 5 Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 3: 9x + 6y - 3z = 33 Add to (2): 2x - y + 3z = 7 Result: 11x + 5y = 40 (Equation A) Step 3: Eliminate z from equations (1) and (3): Multiply…
   Full step-by-step solution

   Step 1: Label the equations: (1) 3x + 2y - z = 11 (2) 2x - y + 3z = 7 (3) x + 4y - 2z = 5 Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 3: 9x + 6y - 3z = 33 Add to (2): 2x - y + 3z = 7 Result: 11x + 5y = 40 (Equation A) Step 3: Eliminate z from equations (1) and (3): Multiply (1) by 2: 6x + 4y - 2z = 22 Subtract (3): x + 4y - 2z = 5 Result: 5x = 17 (Equation B) Step 4: Solve Equation B: x = 17/5 = 3.4 Step 5: Substitute x = 3.4 into Equation A: 11(3.4) + 5y = 40 37.4 + 5y = 40 5y = 2.6 y = 0.52 Step 6: Substitute x = 3.4 and y = 0.52 into equation (1): 3(3.4) + 2(0.52) - z = 11 10.2 + 1.04 - z = 11 11.24 - z = 11 z = 0.24 Step 7: Verify with equation (2): 2(3.4) - 0.52 + 3(0.24) = 6.8 - 0.52 + 0.72 = 7 ✓ Final answer: x = 3.4, y = 0.52, z = 0.24

2. Emma, Liam, and Olivia are partners in a small business that sells handmade crafts. They invested different amounts of money into the business. The total investment from all three partners is $93. Liam invested $3 less than twice Emma's investment. Olivia invested $3 more than the sum of Emma and Liam's investments. Determine how much each person invested. Answer: Emma invested $15, Liam invested $27, Olivia invested $51 Solution: Let x = Emma's investment, y = Liam's investment, z = Olivia's investment. From the total: x + y + z = 93. Liam invested $3 less than twice Emma's: y = 2x - 3.
   Full step-by-step solution

   Step 1: Let x = Emma's investment, y = Liam's investment, z = Olivia's investment. Step 2: From the total: x + y + z = 93. Step 3: Liam invested $3 less than twice Emma's: y = 2x - 3. Step 4: Olivia invested $3 more than the sum of Emma and Liam: z = x + y + 3. Step 5: Substitute y from Step 3 into Step 4: z = x + (2x - 3) + 3 = 3x. Step 6: Substitute y = 2x - 3 and z = 3x into the total equation: x + (2x - 3) + 3x = 93. Step 7: Combine like terms: 6x - 3 = 93. Step 8: Add 3 to both sides: 6x = 96. Step 9: Divide by 6: x = 16. Step 10: Then y = 2(16) - 3 = 32 - 3 = 29. Step 11: And z = 3(16) = 48. Step 12: Check: 16 + 29 + 48 = 93. Correct. Final answer: Emma invested $16, Liam invested $29, Olivia invested $48.

3. 3x + 4y - 2z = 22, 2x - 5y + 3z = -13, 4x + 3y - z = 19 Answer: x = 3, y = 2, z = -1 Solution: (1) 3x + 4y - 2z = 22 (2) 2x - 5y + 3z = -13 (3) 4x + 3y - z = 19 Eliminate z from (1) and (3). Multiply (3) by 2: 8x + 6y - 2z = 38. Subtract (1) from this: (8x + 6y - 2z) - (3x + 4y - 2z) = 38 - 22 → 5x + 2y = 16.
   Full step-by-step solution

   Step 1: Label the equations: (1) 3x + 4y - 2z = 22 (2) 2x - 5y + 3z = -13 (3) 4x + 3y - z = 19 Step 2: Eliminate z from (1) and (3). Multiply (3) by 2: 8x + 6y - 2z = 38. Subtract (1) from this: (8x + 6y - 2z) - (3x + 4y - 2z) = 38 - 22 → 5x + 2y = 16. Call this (A). Step 3: Eliminate z from (2) and (3). Multiply (3) by 3: 12x + 9y - 3z = 57. Add to (2): (2x - 5y + 3z) + (12x + 9y - 3z) = -13 + 57 → 14x + 4y = 44. Divide by 2: 7x + 2y = 22. Call this (B). Step 4: Solve (A) and (B): (A) 5x + 2y = 16 (B) 7x + 2y = 22 Subtract (A) from (B): (7x + 2y) - (5x + 2y) = 22 - 16 → 2x = 6 → x = 3. Step 5: Substitute x = 3 into (A): 5(3) + 2y = 16 → 15 + 2y = 16 → 2y = 1 → y = 1/2. Step 6: Substitute x = 3 and y = 1/2 into (3): 4(3) + 3(1/2) - z = 19 → 12 + 3/2 - z = 19 → 27/2 - z = 19 → -z = 19 - 27/2 = 38/2 - 27/2 = 11/2 → z = -11/2. Step 7: Verify with (1): 3(3) + 4(1/2) - 2(-11/2) = 9 + 2 + 11 = 22 ✓. The solution is x = 3, y = 1/2, z = -11/2.

4. Charlotte is analyzing a three-dimensional geometric structure. A plane defined by the equation 7x + 2y - 2z = 22 intersects the coordinate axes at three points in the first octant (where x ≥ 0, y ≥ 0, z ≥ 0). On a visual diagram, these three intercepts and the origin form a tetrahedron. Determine the coordinates of the three points where the plane meets the x-axis, y-axis, and z-axis. Answer: (22/7, 0, 0), (0, 11, 0), (0, 0, -11) Solution: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 7x + 2(0) - 2(0) = 22, which simplifies to 7x = 22.
   Full step-by-step solution

   Step 1: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 7x + 2(0) - 2(0) = 22, which simplifies to 7x = 22. Divide both sides by 7: x = 22/7. The point is (22/7, 0, 0). Step 2: Find the y-intercept. On the y-axis, x = 0 and z = 0. Substitute into 7(0) + 2y - 2(0) = 22, which simplifies to 2y = 22. Divide both sides by 2: y = 11. The point is (0, 11, 0). Step 3: Find the z-intercept. On the z-axis, x = 0 and y = 0. Substitute into 7(0) + 2(0) - 2z = 22, which simplifies to -2z = 22. Divide both sides by -2: z = -11. The point is (0, 0, -11). The three intersection points are (22/7, 0, 0), (0, 11, 0), and (0, 0, -11).

5. Olivia is analyzing a three-dimensional crystal lattice structure. A plane cuts through the lattice, defined by the equation 3x - 5y + 7z = 21. The plane intersects the x-axis, y-axis, and z-axis at three distinct points in space, forming a triangular cross-section in the first octant. Find the coordinates of these three intersection points. Answer: (7, 0, 0), (0, -21/5, 0), (0, 0, 3) Solution: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 3x - 5(0) + 7(0) = 21, so 3x = 21, x = 7.
   Full step-by-step solution

   Step 1: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 3x - 5(0) + 7(0) = 21, so 3x = 21, x = 7. Point A: (7, 0, 0). Step 2: Find the y-intercept. On the y-axis, x = 0 and z = 0. Substitute: 3(0) - 5y + 7(0) = 21, so -5y = 21, y = -21/5. Point B: (0, -21/5, 0). Step 3: Find the z-intercept. On the z-axis, x = 0 and y = 0. Substitute: 3(0) - 5(0) + 7z = 21, so 7z = 21, z = 3. Point C: (0, 0, 3). The three intersection points are (7, 0, 0), (0, -21/5, 0), and (0, 0, 3).

6. A civil engineering firm is designing a suspension bridge where the main cables follow a parabolic path. The cable's height above the roadway is modeled by the equation h(x) = ax² + bx + c, where x is the horizontal distance from the left tower in meters. Measurements show that at x = 0 (left tower), the cable is 50 meters high; at x = 100 meters (center), the cable is 10 meters high; and at x = 200 meters (right tower), the cable is 50 meters high. Determine the coefficients a, b, and c that define the cable's parabolic path. Answer: a = 0.004, b = -0.8, c = 50 Solution: For a parabola in standard form, substituting the coordinates of three distinct points gives us three equations that can be solved simultaneously to find the coefficients that define the curve's specific shape and position.
   Full step-by-step solution

   When modeling real-world phenomena with quadratic functions, we can use known points to create a system of equations. For a parabola in standard form, substituting the coordinates of three distinct points gives us three equations that can be solved simultaneously to find the coefficients that define the curve's specific shape and position.

7. A triangular prism has vertices at points A(0,0,0), B(4,0,0), C(0,3,0), D(0,0,5), E(4,0,5), and F(0,3,5) in three-dimensional space. Find the volume of this prism. Answer: 30 Solution: The prism has triangular bases and is positioned in 3D space. Points A(0,0,0), B(4,0,0), C(0,3,0) are in the plane z = 0. Points D(0,0,5), E(4,0,5), F(0,3,5) are in the plane z = 5.
   Full step-by-step solution

   Step 1: Understand the shape The prism has triangular bases and is positioned in 3D space. Points A(0,0,0), B(4,0,0), C(0,3,0) are in the plane z = 0. Points D(0,0,5), E(4,0,5), F(0,3,5) are in the plane z = 5. So the triangular base is in the xy-plane at z = 0, and the prism extends in the z-direction by height 5. Step 2: Identify the base triangle The base triangle is ABC: A(0,0,0), B(4,0,0), C(0,3,0). This is a right triangle with legs along the x-axis and y-axis. Step 3: Find the area of the base triangle The base length AB = 4 (from x=0 to x=4). The height AC = 3 (from y=0 to y=3). Area of a right triangle = (1/2) * base * height Area = (1/2) * 4 * 3 = (1/2) * 12 = 6. Step 4: Find the height of the prism The prism height is the perpendicular distance between the two triangular faces. Since the triangles are in planes z = 0 and z = 5, the height is 5. Step 5: Volume of prism Volume = (Area of base) * (height of prism) Volume = 6 * 5 = 30. Step 6: Final answer The volume of the triangular prism is 30.