xy=1 2x+y=8 by equality

To solve this system of equations: 1. xy = 1 2. 2x + y = 8 You can use the method of substitution. First, let's solve one of the equations for one of the variables and then substitute it into the other equation: From equation 1, we can solve for y: xy = 1 y = 1/x Now, substitute this expression for y into equation 2: 2x + (1/x) = 8 To get rid of the fraction, multiply both sides of the equation by x: 2x^2 + 1 = 8x Rearrange the terms: 2x^2 - 8x + 1 = 0 Now, you have a quadratic equation. You can solve it using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a) In this case, a = 2, b = -8, and c = 1. Plug these values into the quadratic formula: x = (-(-8) ± √((-8)^2 - 4(2)(1))) / (2(2)) x = (8 ± √(64 - 8)) / 4 x = (8 ± √56) / 4 Now, simplify: x = (8 ± 2√14) / 4 x = (2(4 ± √14)) / 4 x = (4 ± √14) / 2 Now, you have two possible values for x: 1. x = (4 + √14) / 2 2. x = (4 - √14) / 2 Now, substitute these values back into the equation y = 1/x to find the corresponding values of y: 1. If x = (4 + √14) / 2, then y = 1 / [(4 + √14) / 2] = 2 / (4 + √14) 2. If x = (4 - √14) / 2, then y = 1 / [(4 - √14) / 2] = 2 / (4 - √14) So, there are two sets of solutions for this system of equations: 1. (x, y) = ((4 + √14) / 2, 2 / (4 + √14)) 2. (x, y) = ((4 - √14) / 2, 2 / (4 - √14))