A binary operation is defined by x y = xy. If x * 2 = 12 - x, find the possible values of x.

Here are the solutions to the problems. 21. A binary operation is defined by x y = xy. If x * 2 = 12 - x, find the possible values of x. Step 1: Use the definition of the binary operation. Given x y = xy, we can find x 2 by substituting y=2: x * 2 = x · 2 = 2x Step 2: Set up the equation. We are given that x * 2 = 12 - x. Equating this with our result from Step 1: 2x = 12 - x Step 3: Solve for x. Add x to both sides: 2x + x = 12 3x = 12 Divide by 3: x = (12)/(3) x = 4 Note on options: The literal interpretation of the problem yields a single value x=4. However, the multiple-choice options are pairs of values, suggesting a quadratic equation was intended. A common scenario for this type of question is a typo where x2 was meant to be xx. If we assume the question intended x*x = 12-x: x · x = 12 - x x^2 = 12 - x Rearrange into a standard quadratic form: x^2 + x - 12 = 0 Factor the quadratic equation: (x+4)(x-3) = 0 This gives two possible values for x: x+4 = 0 x = -4 x-3 = 0 x = 3 The values are 3, -4. This matches option C. Given the format of the options, this interpretation is highly probable. The final answer is C. 3, -4. 22. Find y, if 5 & -6 \\ 2 & -7 x \\ y = 7 \\ -11 Step 1: Perform the matrix multiplication. Multiply the 2 × 2 matrix by the 2 × 1 column vector: (5)(x) + (-6)(y) \\ (2)(x) + (-7)(y) = 7 \\ -11 5x - 6y \\ 2x - 7y = 7 \\ -11 Step 2: Form a system of linear equations. Equating the corresponding elements of the matrices gives: 1) 5x - 6y = 7 2) 2x - 7y = -11 Step 3: Solve the system for y using elimination. To eliminate x, multiply equation (1) by 2 and equation (2) by 5: 2 × (5x - 6y = 7) 10x - 12y = 14 (Equation 3) 5 × (2x - 7y = -11) 10x - 35y = -55 (Equation 4)