Solve Systems Exactly

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Solve Systems Exactly · Grade 9 · Worksheet 2

1. Solve the system: 3x² - 2x + 5 = 2x² + 3x - 1 and 4x - y = 7 Answer: x = 2, y = 1 Solution: Simplify the first equation: 3x² - 2x + 5 = 2x² + 3x - 1 Subtract 2x² from both sides: x² - 2x + 5 = 3x - 1 Subtract 3x from both sides: x² - 5x + 5 = -1 Add 1 to both sides: x² - 5x + 6 = 0 Factor the quadratic: (x - 2)(x - 3) = 0 So x = 2 or x = 3 Use the second equation 4x - y = 7 For x = 2:…
   Full step-by-step solution

   Step 1: Simplify the first equation: 3x² - 2x + 5 = 2x² + 3x - 1 Subtract 2x² from both sides: x² - 2x + 5 = 3x - 1 Subtract 3x from both sides: x² - 5x + 5 = -1 Add 1 to both sides: x² - 5x + 6 = 0 Step 2: Factor the quadratic: (x - 2)(x - 3) = 0 So x = 2 or x = 3 Step 3: Use the second equation 4x - y = 7 For x = 2: 4(2) - y = 7 → 8 - y = 7 → y = 1 For x = 3: 4(3) - y = 7 → 12 - y = 7 → y = 5 Step 4: Verify both solutions in the original equations: For (2,1): 3(4) - 4 + 5 = 13 and 2(4) + 6 - 1 = 13 ✓ For (3,5): 3(9) - 6 + 5 = 26 and 2(9) + 9 - 1 = 26 ✓ Both pairs satisfy the system, so the solutions are x = 2, y = 1 and x = 3, y = 5.

2. On a coordinate plane, Noah draws a parabola and a line. The parabola is represented by the equation y = x² - 6x + 11. The line passes through the points (1, 6) and (6, 16). Visualize the graph where the line intersects the parabola at two points. Find the exact coordinates of both intersection points. Answer: (1, 6) and (6, 16) Solution: Find the equation of the line. Slope m = (16 - 6) / (6 - 1) = 10 / 5 = 2. Using point (1, 6): y - 6 = 2(x - 1) => y = 2x + 4.
   Full step-by-step solution

   Step 1: Find the equation of the line. Slope m = (16 - 6) / (6 - 1) = 10 / 5 = 2. Using point (1, 6): y - 6 = 2(x - 1) => y = 2x + 4. Step 2: Set the line equal to the parabola: 2x + 4 = x² - 6x + 11. Step 3: Rearrange to standard form: 0 = x² - 6x + 11 - 2x - 4 => 0 = x² - 8x + 7. Step 4: Factor the quadratic: (x - 1)(x - 7) = 0 => x = 1 or x = 7. Step 5: Find y for each x using y = 2x + 4: when x = 1, y = 2(1) + 4 = 6; when x = 7, y = 2(7) + 4 = 18. Step 6: Verify both points satisfy the parabola: For (1, 6): 1² - 6(1) + 11 = 1 - 6 + 11 = 6. For (7, 18): 7² - 6(7) + 11 = 49 - 42 + 11 = 18. The intersection points are (1, 6) and (7, 18).

3. A chemistry lab needs to mix two solutions with different concentrations of acid. Solution A contains 25% acid and Solution B contains 40% acid. The lab technician wants to create 200 milliliters of a mixture that contains 32% acid. Write and solve a system of equations to determine exactly how many milliliters of each solution should be used. Answer: x = 106.67, y = 93.33 Solution: Let x = milliliters of Solution A (25% acid) and y = milliliters of Solution B (40% acid) Write the volume equation: x + y = 200 Write the acid content equation: 0.25x + 0.40y = 0.32(200) Simplify the acid equation: 0.25x + 0.40y = 64 Solve the system using substitution.
   Full step-by-step solution

   Step 1: Let x = milliliters of Solution A (25% acid) and y = milliliters of Solution B (40% acid) Step 2: Write the volume equation: x + y = 200 Step 3: Write the acid content equation: 0.25x + 0.40y = 0.32(200) Step 4: Simplify the acid equation: 0.25x + 0.40y = 64 Step 5: Solve the system using substitution. From the first equation: y = 200 - x Step 6: Substitute into the acid equation: 0.25x + 0.40(200 - x) = 64 Step 7: Simplify: 0.25x + 80 - 0.40x = 64 Step 8: Combine like terms: -0.15x + 80 = 64 Step 9: Subtract 80 from both sides: -0.15x = -16 Step 10: Divide both sides by -0.15: x = 106.67 Step 11: Substitute back to find y: y = 200 - 106.67 = 93.33 Step 12: The solution is x = 106.67 ml of Solution A and y = 93.33 ml of Solution B

4. Solve the system: x² + y² = 61 and y = x + 1. Answer: x = -6, y = -5 and x = 5, y = 6 Solution: Substitute y = x + 1 into x² + y² = 61. x² + (x + 1)² = 61 → x² + x² + 2x + 1 = 61 → 2x² + 2x + 1 = 61. Subtract 61 from both sides: 2x² + 2x - 60 = 0.
   Full step-by-step solution

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

5. Sophia is a landscape architect designing a new park. The park's main path follows a curve described by the equation y = x² - 7x + 10, where x and y are distances in meters from the entrance. A straight walking trail passes through the park and follows the line y = 2x - 8. At exactly how many meters from the entrance (the x-coordinates) do the main path and the walking trail intersect? Find the exact x-coordinates of all intersection points. Answer: x = 3 and x = 6 Solution: Set the equations equal to each other since at the intersection points, the y-values are the same. x² - 7x + 10 = 2x - 8 Rearrange to get all terms on one side (standard quadratic form).
   Full step-by-step solution

   Step 1: Set the equations equal to each other since at the intersection points, the y-values are the same. x² - 7x + 10 = 2x - 8 Step 2: Rearrange to get all terms on one side (standard quadratic form). x² - 7x + 10 - 2x + 8 = 0 x² - 9x + 18 = 0 Step 3: Factor the quadratic expression. x² - 9x + 18 = 0 We need two numbers that multiply to 18 and add to -9. These numbers are -3 and -6. (x - 3)(x - 6) = 0 Step 4: Apply the zero product property. x - 3 = 0 or x - 6 = 0 x = 3 or x = 6 Step 5: Verify the solutions by substituting them back into both original equations. For x = 3: Main path: y = (3)² - 7(3) + 10 = 9 - 21 + 10 = -2 Walking trail: y = 2(3) - 8 = 6 - 8 = -2 They match at (3, -2). For x = 6: Main path: y = (6)² - 7(6) + 10 = 36 - 42 + 10 = 4 Walking trail: y = 2(6) - 8 = 12 - 8 = 4 They match at (6, 4). The paths intersect at x = 3 meters and x = 6 meters from the entrance.

6. Solve the system: y = x² - 3x + 5 and y = 2x - 1. Answer: x = 2, y = 3 and x = 3, y = 5 Solution: Set the equations equal: x² - 3x + 5 = 2x - 1. Rearrange to standard form: x² - 3x - 2x + 5 + 1 = 0 → x² - 5x + 6 = 0. Factor the quadratic: (x - 2)(x - 3) = 0.
   Full step-by-step solution

   Step 1: Set the equations equal: x² - 3x + 5 = 2x - 1. Step 2: Rearrange to standard form: x² - 3x - 2x + 5 + 1 = 0 → x² - 5x + 6 = 0. Step 3: Factor the quadratic: (x - 2)(x - 3) = 0. Step 4: Solve for x: x = 2 or x = 3. Step 5: Substitute x = 2 into y = 2x - 1: y = 2(2) - 1 = 4 - 1 = 3. Step 6: Substitute x = 3 into y = 2x - 1: y = 2(3) - 1 = 6 - 1 = 5. Step 7: Verify with y = x² - 3x + 5: For x = 2: 4 - 6 + 5 = 3 ✓; For x = 3: 9 - 9 + 5 = 5 ✓. The solutions are x = 2, y = 3 and x = 3, y = 5.

7. A chemistry lab needs to mix two solutions with different concentrations of acid. Solution A contains 15% acid and Solution B contains 40% acid. The lab wants to create 200 milliliters of a mixture that contains exactly 30% acid. Let x represent the amount of Solution A in milliliters and y represent the amount of Solution B in milliliters. Write and solve a system of equations to determine exactly how many milliliters of each solution should be used. Answer: x = 80, y = 120 Solution: When solving mixture problems, we typically set up two equations: one representing the total quantity (like total volume) and another representing the total amount of the substance being mixed (like pure acid).
   Full step-by-step solution

   When solving mixture problems, we typically set up two equations: one representing the total quantity (like total volume) and another representing the total amount of the substance being mixed (like pure acid). The concentration percentage tells us what fraction of each solution is the pure substance. For example, if you had 100 ml of a 20% solution, it would contain 20 ml of pure substance.