Solve Systems Exactly

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Solve Systems Exactly · Grade 9 · Worksheet 3

1. x² + y² = 73, y = 3x + 1. Solve system. Answer: x = -3, y = -8 and x = 3, y = 10 Solution: Substitute y = 3x + 1 into x² + y² = 73.
   Full step-by-step solution

   Step 1: Substitute y = 3x + 1 into x² + y² = 73. Step 2: x² + (3x + 1)² = 73 Step 3: Expand (3x + 1)² = 9x² + 6x + 1 Step 4: x² + 9x² + 6x + 1 = 73 → 10x² + 6x + 1 = 73 Step 5: Subtract 73 from both sides: 10x² + 6x - 72 = 0 Step 6: Divide by 2: 5x² + 3x - 36 = 0 Step 7: Factor: (5x + 12)(x - 3) = 0? Check: 5x * x = 5x², 5x * (-3) = -15x, 12 * x = 12x, 12 * (-3) = -36. Sum: -15x + 12x = -3x. This gives 5x² - 3x - 36 = 0, which is not correct. Let's factor correctly. Step 7 (corrected): Solve 5x² + 3x - 36 = 0. Multiply a*c = 5*(-36) = -180. Find factors of -180 that add to 3: 15 and -12 (since 15 + (-12) = 3). Rewrite: 5x² + 15x - 12x - 36 = 0. Group: 5x(x + 3) - 12(x + 3) = 0 → (5x - 12)(x + 3) = 0. Step 8: So x = -3 or x = 12/5 = 2.4. Step 9: For x = -3, y = 3(-3) + 1 = -9 + 1 = -8. Step 10: For x = 12/5, y = 3(12/5) + 1 = 36/5 + 1 = 36/5 + 5/5 = 41/5 = 8.2. Step 11: Verify: For (-3, -8): (-3)² + (-8)² = 9 + 64 = 73 ✓. For (12/5, 41/5): (144/25) + (1681/25) = 1825/25 = 73 ✓. The solutions are x = -3, y = -8 and x = 12/5, y = 41/5.

2. 2x + 3y = 12 and 3x - 2y = 5 Answer: x = 3, y = 2 Solution: Step 1: Multiply the first equation by 2: 2(2x + 3y) = 2(12) → 4x + 6y = 24 Step 2: Multiply the second equation by 3: 3(3x - 2y) = 3(5) → 9x - 6y = 15 Step 3: Add the two equations: (4x + 6y) + (9x - 6y) = 24 + 15 → 13x = 39 Step 4: Solve for x: x = 39 ÷ 13 = 3 Step 5: Substitute x = 3 into the…
   Full step-by-step solution

   Step 1: Multiply the first equation by 2: 2(2x + 3y) = 2(12) → 4x + 6y = 24 Step 2: Multiply the second equation by 3: 3(3x - 2y) = 3(5) → 9x - 6y = 15 Step 3: Add the two equations: (4x + 6y) + (9x - 6y) = 24 + 15 → 13x = 39 Step 4: Solve for x: x = 39 ÷ 13 = 3 Step 5: Substitute x = 3 into the first equation: 2(3) + 3y = 12 → 6 + 3y = 12 Step 6: Solve for y: 3y = 6 → y = 2 Step 7: Verify with second equation: 3(3) - 2(2) = 9 - 4 = 5 ✓ The solution is x = 3, y = 2.

3. 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 it exactly to find the dimensions of Liam's garden. Answer: width = 6 meters, length = 14 meters Solution: Systems of equations allow us to solve problems with multiple unknown quantities by establishing relationships between them.
   Full step-by-step solution

   Systems of equations allow us to solve problems with multiple unknown quantities by establishing relationships between them. In real-world applications like design and construction, we often know the total measurement (like perimeter) and the proportional relationship between dimensions. By creating equations that model these constraints simultaneously, we can find exact values for all unknowns. This method is widely used in engineering, architecture, and various planning scenarios where multiple conditions must be satisfied at once.

4. Aroha is designing a rectangular garden. The length of the garden is 3 meters more than its width. The area of the garden is 108 square meters. Write and solve a system of equations to find the exact dimensions of the garden. Answer: width = 9 meters, length = 12 meters Solution: Let w represent the width in meters and l represent the length in meters. Step 2: From the problem, the length is 3 meters more than the width: l = w + 3.
   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 3 meters more than the width: l = w + 3. Step 3: The area of a rectangle is length times width: l * w = 108. Step 4: Substitute the expression for l from Step 2 into the area equation: (w + 3) * w = 108. Step 5: Expand: w^2 + 3w = 108. Step 6: Rearrange into standard quadratic form: w^2 + 3w - 108 = 0. Step 7: Factor the quadratic: (w + 12)(w - 9) = 0. Step 8: Set each factor equal to zero: w + 12 = 0 gives w = -12 (not possible for a width), and w - 9 = 0 gives w = 9. Step 9: So the width is 9 meters. Step 10: Find the length: l = w + 3 = 9 + 3 = 12 meters. Step 11: Verify: area = 9 * 12 = 108 square meters. The exact dimensions are width = 9 meters and length = 12 meters.

5. Matiu is designing a rectangular garden. The length of the garden is 6 meters more than twice its width. The area of the garden is 216 square meters. Write and solve a system of equations to find the exact dimensions (width and length) of the garden. Answer: width = 9 meters, length = 24 meters Solution: Let w = width (in meters) and l = length (in meters). From the problem: l = 2w + 6 (length is 6 more than twice the width). Area equation: l * w = 216.
   Full step-by-step solution

   Step 1: Let w = width (in meters) and l = length (in meters). Step 2: From the problem: l = 2w + 6 (length is 6 more than twice the width). Step 3: Area equation: l * w = 216. Step 4: Substitute l from step 2 into step 3: (2w + 6) * w = 216. Step 5: Expand: 2w^2 + 6w = 216. Step 6: Set to zero: 2w^2 + 6w - 216 = 0. Step 7: Divide by 2: w^2 + 3w - 108 = 0. Step 8: Factor: (w + 12)(w - 9) = 0. Step 9: Solve: w = -12 (discard, width cannot be negative) or w = 9. Step 10: Find length: l = 2(9) + 6 = 18 + 6 = 24. Step 11: Check: 9 * 24 = 216 square meters. Correct. The garden is 9 meters wide and 24 meters long.

6. On a coordinate plane, Liam draws a parabola that opens upward with vertex at (3, 1) and a line that passes through the points (1, 5) and (5, 5). The parabola and the line intersect at two points. Visualize this graph. What are the exact coordinates of the two intersection points? Answer: (1, 5) and (5, 5) Solution: The parabola has vertex (3, 1). Its equation in vertex form is y = a(x - 3)^2 + 1. Since it opens upward, a > 0.
   Full step-by-step solution

   Step 1: The parabola has vertex (3, 1). Its equation in vertex form is y = a(x - 3)^2 + 1. Since it opens upward, a > 0. We need to find a. The line passes through (1, 5) and (5, 5), so its slope is (5-5)/(5-1) = 0. The line is horizontal at y = 5. Step 2: The parabola must pass through the same points where it meets the line. Substitute (1, 5) into the parabola equation: 5 = a(1 - 3)^2 + 1 => 5 = a(4) + 1 => 4 = 4a => a = 1. So the parabola is y = (x - 3)^2 + 1. Step 3: Set the equations equal: (x - 3)^2 + 1 = 5 => (x - 3)^2 = 4 => x - 3 = 2 or x - 3 = -2 => x = 5 or x = 1. Step 4: The y-coordinate for both is 5. The intersection points are (1, 5) and (5, 5).

7. A chemical company is mixing two solutions with different concentrations of a compound. Solution A contains 15% of the compound, while Solution B contains 35% of the compound. The chemist needs to create 8 liters of a mixture that contains exactly 25% of the compound. Write and solve a system of equations to determine how many liters of each solution should be used. Answer: 4 liters of Solution A and 4 liters of Solution B Solution: Let x = liters of Solution A, y = liters of Solution B Total volume equation: x + y = 8 Total compound equation: 0.15x + 0.35y = 0.25(8) Simplify compound equation: 0.15x + 0.35y = 2 Solve the system using substitution: y = 8 - x Substitute into compound equation: 0.15x + 0.35(8 - x) = 2 Expand:…
   Full step-by-step solution

   Step 1: Let x = liters of Solution A, y = liters of Solution B Step 2: Total volume equation: x + y = 8 Step 3: Total compound equation: 0.15x + 0.35y = 0.25(8) Step 4: Simplify compound equation: 0.15x + 0.35y = 2 Step 5: Solve the system using substitution: y = 8 - x Step 6: Substitute into compound equation: 0.15x + 0.35(8 - x) = 2 Step 7: Expand: 0.15x + 2.8 - 0.35x = 2 Step 8: Combine like terms: -0.2x + 2.8 = 2 Step 9: Subtract 2.8 from both sides: -0.2x = -0.8 Step 10: Divide by -0.2: x = 4 Step 11: Substitute back: y = 8 - 4 = 4 Step 12: The chemist needs 4 liters of Solution A and 4 liters of Solution B.

8. x² + y² = 61, y = x + 1. Solve system. Answer: x = -6, y = -5 and x = 5, y = 6 Solution: Substitute y = x + 1 into x² + y² = 61.
   Full step-by-step solution

   Step 1: Substitute y = x + 1 into x² + y² = 61. Step 2: x² + (x + 1)² = 61 Step 3: Expand (x + 1)² = x² + 2x + 1 Step 4: x² + x² + 2x + 1 = 61 → 2x² + 2x + 1 = 61 Step 5: Subtract 61 from both sides: 2x² + 2x - 60 = 0 Step 6: Divide by 2: x² + x - 30 = 0 Step 7: Factor: (x + 6)(x - 5) = 0 Step 8: So x = -6 or x = 5 Step 9: For x = -6, y = -6 + 1 = -5 Step 10: For x = 5, y = 5 + 1 = 6 Step 11: Verify: For (-6, -5): (-6)² + (-5)² = 36 + 25 = 61 ✓; For (5, 6): 5² + 6² = 25 + 36 = 61 ✓ The solutions are x = -6, y = -5 and x = 5, y = 6.