{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.