Linear Systems

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Answer Key & Explanations

Linear Systems · Grade 12 · Worksheet 1

1. In a three-dimensional coordinate system, a plane is defined by the equation 5x + 10y - 15z = 30. This plane forms a triangular region in the first octant (where x ≥ 0, y ≥ 0, z ≥ 0) by intersecting the x-axis, y-axis, and z-axis. Emma is studying the geometry of this plane. Find the coordinates of the three points where the plane intersects the axes. Answer: (6, 0, 0), (0, 3, 0), (0, 0, -2) Solution: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 5x + 10y - 15z = 30: 5x + 10(0) - 15(0) = 30, so 5x = 30, x = 6.
   Full step-by-step solution

   Step 1: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 5x + 10y - 15z = 30: 5x + 10(0) - 15(0) = 30, so 5x = 30, x = 6. The point is (6, 0, 0). Step 2: Find the y-intercept. On the y-axis, x = 0 and z = 0. Substitute: 5(0) + 10y - 15(0) = 30, so 10y = 30, y = 3. The point is (0, 3, 0). Step 3: Find the z-intercept. On the z-axis, x = 0 and y = 0. Substitute: 5(0) + 10(0) - 15z = 30, so -15z = 30, z = -2. The point is (0, 0, -2). The three intersection points are (6, 0, 0), (0, 3, 0), and (0, 0, -2).

2. A triangular region in the coordinate plane is bounded by the lines y = 3x - 4, y = -2x + 11, and the x-axis. Find the area of this triangular region using integration. Answer: 12.25 Solution: Find the vertices of the triangle by solving systems of equations. Intersection of y = 3x - 4 and y = -2x + 11: 3x - 4 = -2x + 11 5x = 15 x = 3 y = 3(3) - 4 = 5 Point A: (3, 5) Intersection of y = 3x - 4 and y = 0 (x-axis): 0 = 3x - 4 3x = 4 x = 4/3 Point B: (4/3, 0) Intersection of y = -2x + 11…
   Full step-by-step solution

   Step 1: Find the vertices of the triangle by solving systems of equations. Step 2: Intersection of y = 3x - 4 and y = -2x + 11: 3x - 4 = -2x + 11 5x = 15 x = 3 y = 3(3) - 4 = 5 Point A: (3, 5) Step 3: Intersection of y = 3x - 4 and y = 0 (x-axis): 0 = 3x - 4 3x = 4 x = 4/3 Point B: (4/3, 0) Step 4: Intersection of y = -2x + 11 and y = 0 (x-axis): 0 = -2x + 11 2x = 11 x = 11/2 = 5.5 Point C: (5.5, 0) Step 5: The triangle spans from x = 4/3 to x = 5.5. From x = 4/3 to x = 3, the top boundary is y = 3x - 4. From x = 3 to x = 5.5, the top boundary is y = -2x + 11. Step 6: Calculate the area using integration: Area = ∫ from 4/3 to 3 of (3x - 4) dx + ∫ from 3 to 5.5 of (-2x + 11) dx Step 7: First integral: ∫(3x - 4) dx = (3/2)x² - 4x Evaluate from 4/3 to 3: At x = 3: (3/2)(9) - 4(3) = 13.5 - 12 = 1.5 At x = 4/3: (3/2)(16/9) - 4(4/3) = (3/2)(16/9) - 16/3 = 24/9 - 16/3 = 8/3 - 16/3 = -8/3 First integral = 1.5 - (-8/3) = 1.5 + 8/3 = 3/2 + 8/3 = 9/6 + 16/6 = 25/6 Step 8: Second integral: ∫(-2x + 11) dx = -x² + 11x Evaluate from 3 to 5.5: At x = 5.5: -(5.5)² + 11(5.5) = -30.25 + 60.5 = 30.25 At x = 3: -(9) + 11(3) = -9 + 33 = 24 Second integral = 30.25 - 24 = 6.25 Step 9: Total area = 25/6 + 6.25 = 25/6 + 25/4 = 50/12 + 75/12 = 125/12 = 10.4167 + 1.8333 = 12.25 The answer is 12.25.

3. 2x + 3y - z = 11, x - 2y + 3z = 6, 4x + y - 2z = 15 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 this from the first equation: (2x + 3y - z) - (2x - 4y + 6z) = 11 - 12 → 7y - 7z = -1 → y - z = -1/7 Multiply the second equation by 4: 4(x - 2y + 3z) = 4(6) → 4x - 8y + 12z = 24 Subtract this from the third…
   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 this from the first equation: (2x + 3y - z) - (2x - 4y + 6z) = 11 - 12 → 7y - 7z = -1 → y - z = -1/7 Step 3: Multiply the second equation by 4: 4(x - 2y + 3z) = 4(6) → 4x - 8y + 12z = 24 Step 4: Subtract this from the third equation: (4x + y - 2z) - (4x - 8y + 12z) = 15 - 24 → 9y - 14z = -9 Step 5: Now solve the system: y - z = -1/7 and 9y - 14z = -9 Step 6: Multiply the first equation by 9: 9y - 9z = -9/7 Step 7: Subtract from the second equation: (9y - 14z) - (9y - 9z) = -9 - (-9/7) → -5z = -54/7 → z = 54/35 Step 8: Substitute z = 54/35 into y - z = -1/7 → y - 54/35 = -1/7 → y = 54/35 - 5/35 = 49/35 = 7/5 Step 9: Substitute y and z into the first equation: 2x + 3(7/5) - 54/35 = 11 → 2x + 21/5 - 54/35 = 11 → 2x + 147/35 - 54/35 = 11 → 2x + 93/35 = 11 → 2x = 385/35 - 93/35 = 292/35 → x = 146/35 Step 10: The solution is x = 146/35, y = 7/5, z = 54/35

4. Matiu is analyzing a three-dimensional coordinate system where a plane is defined by the equation 4x - 3y + 2z = 24. The plane intersects the x-axis, y-axis, and z-axis, forming a triangular region in the first octant. Find the coordinates of these three intersection points. Answer: (6, 0, 0), (0, -8, 0), (0, 0, 12) Solution: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 4x - 3(0) + 2(0) = 24, so 4x = 24, x = 6.
   Full step-by-step solution

   Step 1: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 4x - 3(0) + 2(0) = 24, so 4x = 24, x = 6. Point A: (6, 0, 0). Step 2: Find the y-intercept. On the y-axis, x = 0 and z = 0. Substitute: 4(0) - 3y + 2(0) = 24, so -3y = 24, y = -8. Point B: (0, -8, 0). Step 3: Find the z-intercept. On the z-axis, x = 0 and y = 0. Substitute: 4(0) - 3(0) + 2z = 24, so 2z = 24, z = 12. Point C: (0, 0, 12). The three intersection points are (6, 0, 0), (0, -8, 0), and (0, 0, 12).

5. A rectangular prism in three-dimensional space has its vertices at the origin (0,0,0) and at (12, 0, 0), (0, 9, 0), (0, 0, 8), and the corresponding points in the positive x, y, and z directions. A plane cuts through the prism, intersecting the x-axis at x = 10, the y-axis at y = 6, and the z-axis at z = 4. Find the coordinates of the point where the three medians of the triangular region formed by these three intercepts intersect (the centroid of the triangle). Answer: (10/3, 2, 4/3) Solution: Identify the three intercept points. The plane intersects the x-axis at (10, 0, 0), the y-axis at (0, 6, 0), and the z-axis at (0, 0, 4). These are the three vertices of the triangle.
   Full step-by-step solution

   Step 1: Identify the three intercept points. The plane intersects the x-axis at (10, 0, 0), the y-axis at (0, 6, 0), and the z-axis at (0, 0, 4). These are the three vertices of the triangle. Step 2: The centroid of a triangle in 3D space is the average of the coordinates of its three vertices. If the vertices are (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), the centroid is ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3). Step 3: Calculate the x-coordinate of the centroid: (10 + 0 + 0) / 3 = 10/3. Step 4: Calculate the y-coordinate of the centroid: (0 + 6 + 0) / 3 = 6/3 = 2. Step 5: Calculate the z-coordinate of the centroid: (0 + 0 + 4) / 3 = 4/3. Step 6: The centroid is at (10/3, 2, 4/3). The answer is (10/3, 2, 4/3).

6. Hana, Matiu, and Mere are analyzing the budget for their school's cultural festival. The total cost for hiring a sound system, a lighting rig, and a stage is $2100. The cost of the lighting rig is $300 more than the cost of the sound system. The cost of the stage is $600 less than twice the cost of the sound system. Determine the cost of each item: the sound system (x), the lighting rig (y), and the stage (z). Answer: x = 600, y = 900, z = 600 Solution: Define variables: Let x = cost of sound system, y = cost of lighting rig, z = cost of stage. Final answer: Sound system costs $600, lighting rig costs $900, stage costs $600.
   Full step-by-step solution

   Step 1: Define variables: Let x = cost of sound system, y = cost of lighting rig, z = cost of stage. Step 2: Write the equations: Equation 1 (total): x + y + z = 2100 Equation 2 (lighting vs sound): y = x + 300 Equation 3 (stage vs sound): z = 2x - 600 Step 3: Substitute equations 2 and 3 into equation 1: x + (x + 300) + (2x - 600) = 2100 Combine like terms: 4x - 300 = 2100 Add 300 to both sides: 4x = 2400 Divide by 4: x = 600 Step 4: Find y: y = x + 300 = 600 + 300 = 900 Step 5: Find z: z = 2x - 600 = 2(600) - 600 = 1200 - 600 = 600 Step 6: Verify: 600 + 900 + 600 = 2100, and conditions hold. Final answer: Sound system costs $600, lighting rig costs $900, stage costs $600.

7. A three-dimensional coordinate system contains a plane defined by the equation 8x + 9y - 7z = 54. This plane intersects the x-axis, y-axis, and z-axis at three distinct points. Noah is analyzing the triangular region formed by these three intercepts in the first octant. Find the coordinates of these three intersection points. Answer: (27/4, 0, 0), (0, 6, 0), (0, 0, -54/7) Solution: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 8x + 9(0) - 7(0) = 54, so 8x = 54, x = 54/8 = 27/4.
   Full step-by-step solution

   Step 1: Find the x-intercept. On the x-axis, y = 0 and z = 0. Substitute into 8x + 9(0) - 7(0) = 54, so 8x = 54, x = 54/8 = 27/4. Point A: (27/4, 0, 0). Step 2: Find the y-intercept. On the y-axis, x = 0 and z = 0. Substitute: 8(0) + 9y - 7(0) = 54, so 9y = 54, y = 6. Point B: (0, 6, 0). Step 3: Find the z-intercept. On the z-axis, x = 0 and y = 0. Substitute: 8(0) + 9(0) - 7z = 54, so -7z = 54, z = -54/7. Point C: (0, 0, -54/7). The three intersection points are (27/4, 0, 0), (0, 6, 0), and (0, 0, -54/7).