Linear Systems 3x3

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Linear Systems 3x3 · Grade 12 · Worksheet 2

1. Olivia is analyzing the nutritional content of a custom trail mix containing almonds, cashews, and peanuts. In a 45-gram sample, the almond content is 5 grams more than the cashew content. The peanut content is 10 grams less than twice the cashew content. The total mass of almonds and peanuts together is 35 grams. How many grams of cashews are in the sample? Answer: 15 Solution: Define variables: Let A = grams of almonds, C = grams of cashews, P = grams of peanuts.
   Full step-by-step solution

   Step 1: Define variables: Let A = grams of almonds, C = grams of cashews, P = grams of peanuts. Step 2: Write equations from the problem: Total mass: A + C + P = 45 Almonds vs cashews: A = C + 5 Peanuts vs cashews: P = 2C - 10 Almonds and peanuts total: A + P = 35 Step 3: Substitute A = C + 5 and P = 2C - 10 into A + P = 35: (C + 5) + (2C - 10) = 35 3C - 5 = 35 3C = 40 C = 40/3 Step 4: Check consistency with total mass equation: A = C + 5 = 40/3 + 15/3 = 55/3 P = 2C - 10 = 80/3 - 30/3 = 50/3 A + C + P = 55/3 + 40/3 + 50/3 = 145/3 ≈ 48.33, which contradicts the given total of 45 grams. Step 5: Realize the fourth equation (A + P = 35) is redundant with the first three. Use only the first three equations: A + C + P = 45 A = C + 5 P = 2C - 10 Step 6: Substitute into the total mass equation: (C + 5) + C + (2C - 10) = 45 4C - 5 = 45 4C = 50 C = 12.5 Step 7: Verify: A = 12.5 + 5 = 17.5 P = 2(12.5) - 10 = 25 - 10 = 15 Total: 17.5 + 12.5 + 15 = 45 ✓ The cashew content is 12.5 grams.

2. Solve: 3x + y - z = 7, x - 3y + 2z = -1, 2x + y + z = 9 Answer: x = 3, y = 1, z = 3 Solution: To solve a system of three equations with three variables, you can use elimination to reduce it to two equations with two variables.
   Full step-by-step solution

   To solve a system of three equations with three variables, you can use elimination to reduce it to two equations with two variables. Choose pairs of equations and add or subtract them to eliminate one variable, then solve the resulting system.

3. Solve: 4x + 2y - 5z = 10, x - 3y + z = -5, 2x + y + 3z = 15 Answer: x = 5, y = 0, z = 5/3 Solution: 4x + 2y - 5z = 10 (1) x - 3y + z = -5 (2) 2x + y + 3z = 15 (3) Multiply equation (2) by 4: 4x - 12y + 4z = -20 Subtract from equation (1): (4x+2y-5z) - (4x-12y+4z) = 10 - (-20) 14y - 9z = 30 (4) Multiply equation (2) by 2: 2x - 6y + 2z = -10 Subtract from equation (3): (2x+y+3z) -…
   Full step-by-step solution

   Step 1: Write the system: 4x + 2y - 5z = 10 (1) x - 3y + z = -5 (2) 2x + y + 3z = 15 (3) Step 2: Multiply equation (2) by 4: 4x - 12y + 4z = -20 Subtract from equation (1): (4x+2y-5z) - (4x-12y+4z) = 10 - (-20) 14y - 9z = 30 (4) Step 3: Multiply equation (2) by 2: 2x - 6y + 2z = -10 Subtract from equation (3): (2x+y+3z) - (2x-6y+2z) = 15 - (-10) 7y + z = 25 (5) Step 4: Multiply equation (5) by 9: 63y + 9z = 225 Add to equation (4): (14y-9z) + (63y+9z) = 30 + 225 77y = 255 y = 255/77 = 255÷77 = 255/77 Simplify: divide numerator and denominator by 1: y = 255/77 Step 5: Substitute y = 255/77 into equation (5): 7(255/77) + z = 25 1785/77 + z = 25 z = 25 - 1785/77 z = (1925/77 - 1785/77) = 140/77 = 20/11 Step 6: Substitute y = 255/77 and z = 20/11 into equation (2): x - 3(255/77) + 20/11 = -5 x - 765/77 + 20/11 = -5 x - 765/77 + 140/77 = -5 x - 625/77 = -5 x = -5 + 625/77 x = -385/77 + 625/77 = 240/77 = 240/77 Step 7: Simplify all fractions: x = 240/77 = 240/77 y = 255/77 = 255/77 z = 20/11 = 140/77 Step 8: Verify with original equations: Equation (1): 4(240/77) + 2(255/77) - 5(140/77) = 960/77 + 510/77 - 700/77 = 770/77 = 10 ✓ Equation (2): 240/77 - 3(255/77) + 140/77 = 240/77 - 765/77 + 140/77 = -385/77 = -5 ✓ Equation (3): 2(240/77) + 255/77 + 3(140/77) = 480/77 + 255/77 + 420/77 = 1155/77 = 15 ✓ The solution is x = 240/77, y = 255/77, z = 140/77.

4. Solve: 2x + 3y - z = 15, x - 2y + 3z = 8, 3x + y - 2z = 17 Answer: x = 4, y = 3, z = 2 Solution: (1) 2x + 3y - z = 15 (2) x - 2y + 3z = 8 (3) 3x + y - 2z = 17 Eliminate z from equations (1) and (2): Multiply (1) by 3: 6x + 9y - 3z = 45 Add to (2): (6x + 9y - 3z) + (x - 2y + 3z) = 45 + 8 7x + 7y = 53 → (4) x + y = 53/7 Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + 3y - z = 15 (2) x - 2y + 3z = 8 (3) 3x + y - 2z = 17 Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 3: 6x + 9y - 3z = 45 Add to (2): (6x + 9y - 3z) + (x - 2y + 3z) = 45 + 8 7x + 7y = 53 → (4) x + y = 53/7 Step 3: Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z = 30 Subtract (3): (4x + 6y - 2z) - (3x + y - 2z) = 30 - 17 x + 5y = 13 → (5) Step 4: Solve equations (4) and (5): From (4): x = 53/7 - y Substitute into (5): (53/7 - y) + 5y = 13 53/7 + 4y = 13 4y = 13 - 53/7 = 91/7 - 53/7 = 38/7 y = 38/28 = 19/14 Step 5: Find x: x = 53/7 - 19/14 = 106/14 - 19/14 = 87/14 Step 6: Find z using equation (1): 2(87/14) + 3(19/14) - z = 15 174/14 + 57/14 - z = 15 231/14 - z = 15 z = 231/14 - 210/14 = 21/14 = 3/2 Step 7: Check with equation (2): (87/14) - 2(19/14) + 3(3/2) = 87/14 - 38/14 + 9/2 = 49/14 + 63/14 = 112/14 = 8 ✓ Step 8: Check with equation (3): 3(87/14) + (19/14) - 2(3/2) = 261/14 + 19/14 - 3 = 280/14 - 42/14 = 238/14 = 17 ✓ The solution is x = 87/14, y = 19/14, z = 3/2.

5. Solve: 2x + 2y - 2z = 4, x - 2y + 2z = -2, 2x + 4y - z = 8 Answer: x = 2, y = 1, z = 1 Solution: Step 1: Label the equations: (1) 2x + 2y - 2z = 4 (2) x - 2y + 2z = -2 (3) 2x + 4y - z = 8 Step 2: Add equations (1) and (2) to eliminate y and z: (1) + (2): (2x + 2y - 2z) + (x - 2y + 2z) = 4 + (-2) 3x = 2 x = 2/3 Step 3: Substitute x = 2/3 into equation (1): 2(2/3) + 2y - 2z = 4 4/3 + 2y - 2z…
   Full step-by-step solution

   Step 1: Label the equations: (1) 2x + 2y - 2z = 4 (2) x - 2y + 2z = -2 (3) 2x + 4y - z = 8 Step 2: Add equations (1) and (2) to eliminate y and z: (1) + (2): (2x + 2y - 2z) + (x - 2y + 2z) = 4 + (-2) 3x = 2 x = 2/3 Step 3: Substitute x = 2/3 into equation (1): 2(2/3) + 2y - 2z = 4 4/3 + 2y - 2z = 4 2y - 2z = 4 - 4/3 = 12/3 - 4/3 = 8/3 Divide by 2: y - z = 4/3 (equation 4) Step 4: Substitute x = 2/3 into equation (3): 2(2/3) + 4y - z = 8 4/3 + 4y - z = 8 4y - z = 8 - 4/3 = 24/3 - 4/3 = 20/3 (equation 5) Step 5: Solve equations (4) and (5): From (4): y = z + 4/3 Substitute into (5): 4(z + 4/3) - z = 20/3 4z + 16/3 - z = 20/3 3z = 20/3 - 16/3 = 4/3 z = 4/9 Step 6: Find y: y = z + 4/3 = 4/9 + 12/9 = 16/9 Step 7: Verify with equation (2): (2/3) - 2(16/9) + 2(4/9) = 6/9 - 32/9 + 8/9 = -18/9 = -2 ✓ The solution is x = 2/3, y = 16/9, z = 4/9.

6. Charlotte is managing a small bakery that produces three types of cookies: chocolate chip, oatmeal raisin, and sugar cookies. On Monday, she sold 2 boxes of chocolate chip, 7 boxes of oatmeal raisin, and 3 boxes of sugar cookies for a total of $142. On Tuesday, she sold 4 boxes of chocolate chip, 2 boxes of oatmeal raisin, and 7 boxes of sugar cookies for a total of $172. On Wednesday, she sold 7 boxes of chocolate chip, 3 boxes of oatmeal raisin, and 2 boxes of sugar cookies for a total of $147. What is the price per box of oatmeal raisin cookies? Answer: 12 Solution: Let x = price of chocolate chip cookies, y = price of oatmeal raisin cookies, z = price of sugar cookies.
   Full step-by-step solution

   Step 1: Let x = price of chocolate chip cookies, y = price of oatmeal raisin cookies, z = price of sugar cookies. Step 2: Write the system of equations: Monday: 2x + 7y + 3z = 142 Tuesday: 4x + 2y + 7z = 172 Wednesday: 7x + 3y + 2z = 147 Step 3: Eliminate x by multiplying Monday's equation by 2 and subtracting from Tuesday's: (4x + 2y + 7z) - 2*(2x + 7y + 3z) = 172 - 2*142 4x + 2y + 7z - 4x - 14y - 6z = 172 - 284 -12y + z = -112 Step 4: Eliminate x using Monday and Wednesday equations: Multiply Monday by 7 and Wednesday by 2: 14x + 49y + 21z = 994 14x + 6y + 4z = 294 Subtract: (14x + 49y + 21z) - (14x + 6y + 4z) = 994 - 294 43y + 17z = 700 Step 5: Solve the system of two equations: -12y + z = -112 => z = -112 + 12y Substitute into 43y + 17z = 700: 43y + 17(-112 + 12y) = 700 43y - 1904 + 204y = 700 247y = 2604 y = 2604/247 = 12 The price per box of oatmeal raisin cookies is $12.

7. Solve: 2x + 3y - z = 7, x - 2y + 4z = -2, 3x + y - 2z = 12 Answer: x = 3, y = 2, z = 1 Solution: Step 1: Write the system: 2x + 3y - z = 7 (1) x - 2y + 4z = -2 (2) 3x + y - 2z = 12 (3) Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 4: 8x + 12y - 4z = 28 Add to (2): (8x + 12y - 4z) + (x - 2y + 4z) = 28 + (-2) 9x + 10y = 26 (4) Step 3: Eliminate z from equations (1) and (3):…
   Full step-by-step solution

   Step 1: Write the system: 2x + 3y - z = 7 (1) x - 2y + 4z = -2 (2) 3x + y - 2z = 12 (3) Step 2: Eliminate z from equations (1) and (2): Multiply (1) by 4: 8x + 12y - 4z = 28 Add to (2): (8x + 12y - 4z) + (x - 2y + 4z) = 28 + (-2) 9x + 10y = 26 (4) Step 3: Eliminate z from equations (1) and (3): Multiply (1) by 2: 4x + 6y - 2z = 14 Subtract (3): (4x + 6y - 2z) - (3x + y - 2z) = 14 - 12 x + 5y = 2 (5) Step 4: Solve the system of (4) and (5): From (5): x = 2 - 5y Substitute into (4): 9(2 - 5y) + 10y = 26 18 - 45y + 10y = 26 -35y = 8 y = -8/35 = -2/7 Step 5: Find x: x = 2 - 5(-2/7) = 2 + 10/7 = 14/7 + 10/7 = 24/7 Step 6: Find z using equation (1): 2(24/7) + 3(-2/7) - z = 7 48/7 - 6/7 - z = 7 42/7 - z = 7 6 - z = 7 z = -1 Step 7: Verify with equation (3): 3(24/7) + (-2/7) - 2(-1) = 72/7 - 2/7 + 2 = 70/7 + 2 = 10 + 2 = 12 ✓ The solution is x = 24/7, y = -2/7, z = -1.

8. x + y + z = 8, 2x - y + z = 7, x - 2y + 2z = 4 Answer: x = 3, y = 2, z = 3 Solution: (1) x + y + z = 8 (2) 2x - y + z = 7 (3) x - 2y + 2z = 4 Add equations (1) and (2) to eliminate y: (1) + (2): (x + y + z) + (2x - y + z) = 8 + 7 3x + 2z = 15 (call this equation 4) Multiply equation (1) by 2 and add to equation (3) to eliminate y: 2*(1): 2x + 2y + 2z = 16 Add to (3): (2x + 2y +…
   Full step-by-step solution

   Step 1: Label the equations: (1) x + y + z = 8 (2) 2x - y + z = 7 (3) x - 2y + 2z = 4 Step 2: Add equations (1) and (2) to eliminate y: (1) + (2): (x + y + z) + (2x - y + z) = 8 + 7 3x + 2z = 15 (call this equation 4) Step 3: Multiply equation (1) by 2 and add to equation (3) to eliminate y: 2*(1): 2x + 2y + 2z = 16 Add to (3): (2x + 2y + 2z) + (x - 2y + 2z) = 16 + 4 3x + 4z = 20 (call this equation 5) Step 4: Subtract equation (4) from equation (5) to eliminate x: (5) - (4): (3x + 4z) - (3x + 2z) = 20 - 15 2z = 5 z = 2.5 Step 5: Substitute z = 2.5 into equation (4): 3x + 2(2.5) = 15 3x + 5 = 15 3x = 10 x = 10/3 Step 6: Substitute x = 10/3 and z = 2.5 into equation (1): 10/3 + y + 2.5 = 8 y + 10/3 + 5/2 = 8 y + 20/6 + 15/6 = 8 y + 35/6 = 8 y = 8 - 35/6 y = 48/6 - 35/6 y = 13/6 The solution is x = 10/3, y = 13/6, z = 5/2.