{4x - y=2 {x^2 + y^2 =3

To solve the system of equations given by: 1. \( 4x - y = 2 \) 2. \( x^2 + y^2 = 3 \) we can follow these steps: ### Step 1: Solve the first equation for \( y \). From the first equation, we can express \( y \) in terms of \( x \): \[ y = 4x - 2 \] ### Step 2: Substitute \( y \) into the second equation. Now, we substitute \( y \) from the first equation into the second equation: \[ x^2 + (4x - 2)^2 = 3 \] ### Step 3: Expand and simplify. Expanding \( (4x - 2)^2 \): \[ (4x - 2)^2 = 16x^2 - 16x + 4 \] Now, substitute this into the second equation: \[ x^2 + 16x^2 - 16x + 4 = 3 \] Combine like terms: \[ 17x^2 - 16x + 4 = 3 \] Subtract 3 from both sides: \[ 17x^2 - 16x + 1 = 0 \] ### Step 4: Use the quadratic formula. To solve the quadratic equation \( 17x^2 - 16x + 1 = 0 \), we apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 17 \), \( b = -16 \), and \( c = 1 \): \[ b^2 - 4ac = (-16)^2 - 4 \cdot 17 \cdot 1 = 256 - 68 = 188 \] Now, substitute values into the quadratic formula: \[ x = \frac{16 \pm \sqrt{188}}{34} \] ### Step 5: Simplify \( \sqrt{188} \). Finding \( \sqrt{188} \): \[ \sqrt{188} = \sqrt{4 \cdot 47} = 2\sqrt{47} \] So our expression for \( x \) becomes: \[ x = \frac{16 \pm 2\sqrt{47}}{34} = \frac{8 \pm \sqrt{47}}{17} \] ### Step 6: Find corresponding \( y \) values. Now substitute back to find \( y \) values using \( y = 4x - 2 \): 1. For \( x = \frac{8 + \sqrt{47}}{17} \): \[ y = 4\left(\frac{8 + \sqrt{47}}{17}\right) - 2 = \frac{32 + 4\sqrt{47}}{17} - 2 = \frac{32 + 4\sqrt{47} - 34}{17} = \frac{-2 + 4\sqrt{47}}{17} \] 2. For \( x = \frac{8 - \sqrt{47}}{17} \): \[ y = 4\left(\frac{8 - \sqrt{47}}{17}\right) - 2 = \frac{32 - 4\sqrt{47}}{17} - 2 = \frac{32 - 4\sqrt{47} - 34}{17} = \frac{-2 - 4\sqrt{47}}{17} \] ### Step 7: Final solutions. Thus, the solutions to the system of equations are: 1. \( \left( \frac{8 + \sqrt{47}}{17}, \frac{-2 + 4\sqrt{47}}{17} \right) \) 2. \( \left( \frac{8 - \sqrt{47}}{17}, \frac{-2 - 4\sqrt{47}}{17} \right) \) These are the points of intersection of the line and the circle defined by the given equations.