Linear Systems Worksheets Grade 12

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.
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.
3. 2x + 3y - z = 11, x - 2y + 3z = 6, 4x + y - 2z = 15

…and 4 more problems

1. 3x + 2y - z = 11, 2x - y + 3z = 7, x + 4y - 2z = 5
2. Emma, Liam, and Olivia are partners in a small business that sells handmade crafts. They invested different amounts of money into the business. The total investment from all three partners is $93. Liam invested $3 less than twice Emma's investment. Olivia invested $3 more than the sum of Emma and Liam's investments. Determine how much each person invested.
3. 3x + 4y - 2z = 22, 2x - 5y + 3z = -13, 4x + 3y - z = 19

…and 4 more problems

1. A chemical engineering company is optimizing a reactor that uses three catalysts (X, Y, Z) in different trials. In Trial 1, using 3 units of X, 2 units of Y, and 1 unit of Z produced 34 grams of product. In Trial 2, using 1 unit of X, 4 units of Y, and 2 units of Z produced 36 grams. In Trial 3, using 2 units of X, 1 unit of Y, and 3 units of Z produced 35 grams. The effectiveness of each catalyst can be represented by coefficients a, b, and c respectively in the equation: (catalyst amount) × (catalyst effectiveness) = product yield. Determine the effectiveness coefficient for each catalyst.
2. Aroha, Tane, and Kaia are three friends who run a small business selling handmade candles in three different scents: lavender, vanilla, and cinnamon. They sell their candles in gift boxes. For one gift box, the number of lavender candles is three times the number of vanilla candles, and the number of cinnamon candles is 7 more than the number of vanilla candles. The total number of candles in the box is 47. Determine how many of each scent are in the gift box.
3. A chemical engineering company is designing a reactor system where three substances (A, B, and C) must be mixed in specific ratios. Substance A contains 2 units of compound X and 1 unit of compound Y per liter. Substance B contains 1 unit of compound X, 3 units of compound Y, and 2 units of compound Z per liter. Substance C contains 4 units of compound X, 1 unit of compound Y, and 1 unit of compound Z per liter. To achieve the optimal reaction, the final mixture requires exactly 14 units of compound X, 9 units of compound Y, and 5 units of compound Z. How many liters of each substance (A, B, and C) should be used?

…and 2 more problems