This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Substitute the equation of the line into the equation of the curve. Given the curve xy = 12 and the line y = 7 - x. Substitute y = 7 - x into the curve equation: x(7 - x) = 12 Step 2: Expand and rearrange the equation into a quadratic form. 7x - x^2 = 12 Move all terms to one side to get a standard quadratic equation: x^2 - 7x + 12 = 0 Step 3: Solve the quadratic equation for x. Factor the quadratic equation: We need two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4. (x - 3)(x - 4) = 0 This gives two possible values for x: x - 3 = 0 x_1 = 3 x - 4 = 0 x_2 = 4 Step 4: Substitute the values of x back into the linear equation y = 7 - x to find the corresponding y values. For x_1 = 3: y_1 = 7 - 3 y_1 = 4 The first point of intersection is (3, 4). For x_2 = 4: y_2 = 7 - 4 y_2 = 3 The second point of intersection is (4, 3). The coordinates of the two points are (3, 4) and (4, 3).