Translate the following statements into algebraic equations and solve for x and y: x is the sum of 2 and y. Five times the product of x and y, is 6 more than the square of x.

Morning Maths — let's get this done. Step 1: Translate the given statements into algebraic equations. The first statement says "x is the sum of 2 and y". This can be written as: x = 2 + y (1) The second statement says "Five times the product of x and y, is 6 more than the square of x". This can be written as: 5xy = x^2 + 6 (2) Step 2: Substitute equation (1) into equation (2). From equation (1), we can express y in terms of x: y = x - 2 Now, substitute this expression for y into equation (2): 5x(x - 2) = x^2 + 6 Step 3: Solve the resulting quadratic equation for x. Expand the left side of the equation: 5x^2 - 10x = x^2 + 6 Rearrange the terms to form a standard quadratic equation ax^2 + bx + c = 0: 5x^2 - x^2 - 10x - 6 = 0 4x^2 - 10x - 6 = 0 Divide the entire equation by 2 to simplify: 2x^2 - 5x - 3 = 0 Factor the quadratic equation: (2x + 1)(x - 3) = 0 This gives two possible values for x: 2x + 1 = 0 2x = -1 x = -(1)/(2) x - 3 = 0 x = 3 Step 4: Find the corresponding values of y for each value of x. Using the relation y = x - 2: Case 1: For x = 3 y = 3 - 2 y = 1 Case 2: For x = -(1)/(2) y = -(1)/(2) - 2 y = -(1)/(2) - (4)/(2) y = -(5)/(2) The values of x and y are: For the first solution: x = 3, y = 1 For the second solution: x = -(1)/(2), y = -(5)/(2) Send me the next one 📸