Linear Systems Worksheets Grade 12

1. A chemistry lab is conducting three experiments to determine the heat absorption rates (in J/g) of three unknown compounds: X, Y, and Z. In Experiment 1, a sample containing 1 gram of X, 3 grams of Y, and 1 gram of Z absorbs 31 J of heat. In Experiment 2, a sample containing 2 grams of X, 1 gram of Y, and 6 grams of Z absorbs 46 J of heat. In Experiment 3, a sample containing 1 gram of X, 1 gram of Y, and 1 gram of Z absorbs 16 J of heat. If the total heat absorbed is the sum of the heat absorbed by each compound individually (mass × heat absorption rate), determine the heat absorption rate (in J/g) for each compound.
2. A chemical engineering company is designing a reactor system where three substances (A, B, and C) must be mixed in precise ratios. The concentration equations are: 2x + 3y - z = 15 for substance A, x - 2y + 4z = 10 for substance B, and 3x + y - 2z = 8 for substance C. What are the exact concentrations (x, y, z) that satisfy all three equations simultaneously?
3. Solve the system: 2x + y - z = 8, x - 3y + 2z = -1, 3x + 2y - 4z = 9

…and 3 more problems

1. A civil engineering firm is designing a suspension bridge where the main cables follow a parabolic path. The cable's height above the roadway is modeled by the equation h(x) = ax² + bx + c, where x is the horizontal distance from the left tower in meters. Measurements show that at x = 0 (left tower), the cable is 50 meters high; at x = 100 meters (center), the cable is 10 meters high; and at x = 200 meters (right tower), the cable is 50 meters high. Determine the coefficients a, b, and c that define the cable's parabolic path.
2. A chemical engineering company is designing a reactor system with three interconnecting tanks. Tank A receives a solution containing x grams of compound P, y grams of compound Q, and z grams of compound R. Tank B receives a mixture where the amount of compound P is twice that of compound Q, and the amount of compound R is 5 grams more than compound P. Tank C receives a mixture where the total mass is 45 grams, with compound P having 3 grams less than compound Q, and compound R having twice the mass of compound P. If the total mass across all three tanks is 120 grams, and the total amount of compound P is equal to the total amount of compound Q, determine the mass of each compound in Tank A.
3. Isabella, Mason, and Charlotte are partners in a landscaping business. They invested different amounts of money and agreed to share the annual profit proportionally to their investments. Their total investment was $84,000. Mason invested $9,000 more than Isabella, and Charlotte invested $12,000 less than twice Mason's investment. Determine how much each person invested.

…and 3 more problems

1. Solve the system: 2x + y - z = 7, x - 3y + 2z = -1, 3x + 2y - 4z = 4
2. Emma, Liam, and Olivia are managing a small business that sells three types of handcrafted candles: lavender, vanilla, and cinnamon. Last month, the total revenue from all candles was $185. The revenue from lavender candles was $15 more than the revenue from vanilla candles, and the revenue from cinnamon candles was $5 less than twice the revenue from vanilla candles. Determine the revenue from each type of candle.
3. 3x + 5y - 7z = 15, 5x - 3y + 9z = 23, 7x + 9y - 5z = 31

…and 3 more problems