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Systems of Linear Equations

Grade 10 · Mathematics · Worksheet 1

  1. On a coordinate plane, Noah draws a trapezoid with vertices at A(7, 2), B(19, 2), C(15, 10), and D(9, 10). He then draws a diagonal from A to C. A second line passes through the midpoint of this diagonal and is perpendicular to side AB. Visualize this figure. Write the system of linear equations representing the two lines (the diagonal AC and the line through the midpoint perpendicular to AB), then find their intersection point. Answer: ______________
  2. A technology company is analyzing the optimal production mix for two new electronic devices. Device X requires 3 hours of assembly and 1 hour of quality testing, while Device Y requires 2 hours of assembly and 2 hours of quality testing. The factory has exactly 240 assembly hours and 160 quality testing hours available per week. If the company wants to use all available hours exactly, how many units of each device should they produce? Answer: ______________
  3. 2x + 3y = 19; 5x - 2y = 26; x = ? Answer: ______________
  4. Isabella is constructing a geometric figure on a coordinate plane. She plots a triangle with vertices at A(9, 12), B(27, 12), and C(18, 36). She then draws the line that passes through the midpoint of side AB and is parallel to side AC. She also draws the line that passes through the midpoint of side BC and is parallel to side AB. Visualize this triangle and these two lines. Write the system of linear equations representing these two lines, and find the coordinates of their intersection point. Answer: ______________
  5. Isabella is designing a geometric logo on a coordinate plane. She draws a triangle with vertices at A(8, 15), B(20, 15), and C(14, 31). She then draws the median from vertex C to side AB, and the line that passes through the midpoint of median CM and is parallel to side BC. Visualize this figure. Write the system of linear equations representing the median from C to AB and the line through the midpoint of the median parallel to BC, then find the coordinates of their intersection point. Answer: ______________
  6. 3x + 2y = 18; 2x - 3y = -5; x = ? Answer: ______________
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Answer Key & Explanations

Systems of Linear Equations · Grade 10 · Worksheet 1

  1. On a coordinate plane, Noah draws a trapezoid with vertices at A(7, 2), B(19, 2), C(15, 10), and D(9, 10). He then draws a diagonal from A to C. A second line passes through the midpoint of this diagonal and is perpendicular to side AB. Visualize this figure. Write the system of linear equations representing the two lines (the diagonal AC and the line through the midpoint perpendicular to AB), then find their intersection point. Answer: (11, 6) Solution: Find the equation of diagonal AC. A(7, 2) and C(15, 10). Slope m1 = (10 - 2) / (15 - 7) = 8/8 = 1.
    Full step-by-step solution

    Step 1: Find the equation of diagonal AC. A(7, 2) and C(15, 10). Slope m1 = (10 - 2) / (15 - 7) = 8/8 = 1. Using point-slope form with A(7, 2): y - 2 = 1(x - 7) => y = x - 5. (Equation 1) Step 2: Find midpoint M of AC. M = ((7+15)/2, (2+10)/2) = (22/2, 12/2) = (11, 6). Step 3: Find slope of AB. A(7, 2) and B(19, 2). Slope m2 = (2 - 2) / (19 - 7) = 0/12 = 0. So AB is horizontal. A line perpendicular to AB is vertical, so its equation is x = 11. (Equation 2) Step 4: Solve the system: y = x - 5 and x = 11. Substitute x = 11 into y = x - 5: y = 11 - 5 = 6. Step 5: The intersection point is (11, 6). This is exactly the midpoint M, so the two lines intersect at the midpoint of diagonal AC. The answer is (11, 6).

  2. A technology company is analyzing the optimal production mix for two new electronic devices. Device X requires 3 hours of assembly and 1 hour of quality testing, while Device Y requires 2 hours of assembly and 2 hours of quality testing. The factory has exactly 240 assembly hours and 160 quality testing hours available per week. If the company wants to use all available hours exactly, how many units of each device should they produce? Answer: (40,60) Solution: Let x be the number of Device X units and y be the number of Device Y units. Write the equation for assembly hours: 3x + 2y = 240 Write the equation for testing hours: 1x + 2y = 160 Subtract the second equation from the first: (3x + 2y) - (x + 2y) = 240 - 160 Simplify: 2x = 80 Solve for x: x =…
    Full step-by-step solution

    Step 1: Let x be the number of Device X units and y be the number of Device Y units. Step 2: Write the equation for assembly hours: 3x + 2y = 240 Step 3: Write the equation for testing hours: 1x + 2y = 160 Step 4: Subtract the second equation from the first: (3x + 2y) - (x + 2y) = 240 - 160 Step 5: Simplify: 2x = 80 Step 6: Solve for x: x = 40 Step 7: Substitute x = 40 into the second equation: 40 + 2y = 160 Step 8: Solve for y: 2y = 120, y = 60 Step 9: The solution is 40 units of Device X and 60 units of Device Y. The answer is (40,60).

  3. 2x + 3y = 19; 5x - 2y = 26; x = ? Answer: 5 Solution: Multiply the first equation by 2: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Multiply the second equation by 3: 3(5x - 2y) = 3(26) → 15x - 6y = 78 Add the two new equations: (4x + 6y) + (15x - 6y) = 38 + 78 → 19x = 116 Solve for x: x = 116 ÷ 19 = 5 The answer is x = 5.
    Full step-by-step solution

    Step 1: Multiply the first equation by 2: 2(2x + 3y) = 2(19) → 4x + 6y = 38 Step 2: Multiply the second equation by 3: 3(5x - 2y) = 3(26) → 15x - 6y = 78 Step 3: Add the two new equations: (4x + 6y) + (15x - 6y) = 38 + 78 → 19x = 116 Step 4: Solve for x: x = 116 ÷ 19 = 5 The answer is x = 5.

  4. Isabella is constructing a geometric figure on a coordinate plane. She plots a triangle with vertices at A(9, 12), B(27, 12), and C(18, 36). She then draws the line that passes through the midpoint of side AB and is parallel to side AC. She also draws the line that passes through the midpoint of side BC and is parallel to side AB. Visualize this triangle and these two lines. Write the system of linear equations representing these two lines, and find the coordinates of their intersection point. Answer: (18, 24) Solution: Find the midpoint of side AB. A(9, 12) and B(27, 12). Midpoint M_AB = ((9+27)/2, (12+12)/2) = (36/2, 24/2) = (18, 12).
    Full step-by-step solution

    Step 1: Find the midpoint of side AB. A(9, 12) and B(27, 12). Midpoint M_AB = ((9+27)/2, (12+12)/2) = (36/2, 24/2) = (18, 12). Step 2: Find the slope of side AC. A(9, 12) and C(18, 36). Slope m_AC = (36 - 12) / (18 - 9) = 24/9 = 8/3. The first line passes through M_AB(18, 12) and is parallel to AC, so it has slope 8/3. Equation: y - 12 = (8/3)(x - 18). Multiply by 3: 3y - 36 = 8x - 144 => 3y = 8x - 108 => y = (8/3)x - 36. (Equation 1) Step 3: Find the midpoint of side BC. B(27, 12) and C(18, 36). Midpoint M_BC = ((27+18)/2, (12+36)/2) = (45/2, 48/2) = (22.5, 24). Step 4: Find the slope of side AB. A(9, 12) and B(27, 12). Slope m_AB = (12 - 12) / (27 - 9) = 0/18 = 0. The second line passes through M_BC(22.5, 24) and is parallel to AB, so it has slope 0 (horizontal line). Equation: y = 24. (Equation 2) Step 5: Solve the system: y = (8/3)x - 36 and y = 24. Substitute y = 24 into Equation 1: 24 = (8/3)x - 36. Add 36 to both sides: 60 = (8/3)x. Multiply by 3/8: x = 60 * (3/8) = 180/8 = 22.5. Step 6: The intersection point is (22.5, 24). However, note that M_BC is (22.5, 24), so the intersection is exactly at the midpoint of BC. The answer is (22.5, 24).

  5. Isabella is designing a geometric logo on a coordinate plane. She draws a triangle with vertices at A(8, 15), B(20, 15), and C(14, 31). She then draws the median from vertex C to side AB, and the line that passes through the midpoint of median CM and is parallel to side BC. Visualize this figure. Write the system of linear equations representing the median from C to AB and the line through the midpoint of the median parallel to BC, then find the coordinates of their intersection point. Answer: (14, 23) Solution: Find the midpoint M of AB. A(8, 15) and B(20, 15). M = ((8+20)/2, (15+15)/2) = (28/2, 30/2) = (14, 15).
    Full step-by-step solution

    Step 1: Find the midpoint M of AB. A(8, 15) and B(20, 15). M = ((8+20)/2, (15+15)/2) = (28/2, 30/2) = (14, 15). Step 2: Find the equation of median CM, which passes through C(14, 31) and M(14, 15). Since both x-coordinates are 14, this is a vertical line: x = 14. (Equation 1) Step 3: Find the midpoint P of median CM. C(14, 31) and M(14, 15). P = ((14+14)/2, (31+15)/2) = (28/2, 46/2) = (14, 23). Step 4: Find the slope of side BC. B(20, 15) and C(14, 31). Slope = (31 - 15) / (14 - 20) = 16 / (-6) = -8/3. Step 5: The line through P parallel to BC has the same slope -8/3. Using point-slope form with P(14, 23): y - 23 = (-8/3)(x - 14). Multiply both sides by 3: 3y - 69 = -8(x - 14) => 3y - 69 = -8x + 112 => 8x + 3y = 181. (Equation 2) Step 6: Solve the system: x = 14 and 8x + 3y = 181. Substitute x = 14 into the second equation: 8(14) + 3y = 181 => 112 + 3y = 181 => 3y = 69 => y = 23. Step 7: The intersection point is (14, 23), which is exactly the midpoint P of the median. The answer is (14, 23).

  6. 3x + 2y = 18; 2x - 3y = -5; x = ? Answer: 4 Solution: Multiply the first equation by 3: 3(3x + 2y) = 3(18) → 9x + 6y = 54 Multiply the second equation by 2: 2(2x - 3y) = 2(-5) → 4x - 6y = -10 Add the two equations: (9x + 6y) + (4x - 6y) = 54 + (-10) → 13x = 44 Solve for x: x = 44/13 = 4 The answer is x = 4.
    Full step-by-step solution

    Step 1: Multiply the first equation by 3: 3(3x + 2y) = 3(18) → 9x + 6y = 54 Step 2: Multiply the second equation by 2: 2(2x - 3y) = 2(-5) → 4x - 6y = -10 Step 3: Add the two equations: (9x + 6y) + (4x - 6y) = 54 + (-10) → 13x = 44 Step 4: Solve for x: x = 44/13 = 4 The answer is x = 4.