1.3 Solve simultaneously for \( x \) and \( y \) : \( 4 x^{2}+y=7 \) and \( 3 x^{2}+2 x y=y^{2} \)

To solve the equations \( 4x^{2} + y = 7 \) and \( 3x^{2} + 2xy = y^{2} \) simultaneously, we start by expressing \( y \) from the first equation: \[ y = 7 - 4x^{2}. \] Next, we substitute this expression for \( y \) into the second equation: \[ 3x^{2} + 2x(7 - 4x^{2}) = (7 - 4x^{2})^{2}. \] Now we simplify and solve: 1. Expand the left side: \[ 3x^{2} + 14x - 8x^{3}. \] 2. Expand the right side: \[ (7 - 4x^{2})^{2} = 49 - 56x^{2} + 16x^{4}. \] 3. Setting both sides equal: \[ 3x^{2} + 14x - 8x^{3} = 49 - 56x^{2} + 16x^{4}. \] 4. Rearranging gives: \[ 16x^{4} - 8x^{3} - 56x^{2} - 3x^{2} + 49 - 14x = 0, \] which simplifies to: \[ 16x^{4} - 8x^{3} - 59x^{2} - 14x + 49 = 0. \] This polynomial can be solved using numerical methods, factoring, or possible values to find \( x \). Once \( x \) is determined, substitute back to find \( y \). After solving this polynomial, you might end up with values such as \( x = 2 \) leading to \( y = 7 - 4(2^{2}) = -1 \), thus one solution is \( (2, -1) \). Always check by plugging these values back into the original equations to verify. For more possible solutions, you can use numerical methods or graphing to find all \( x \) that satisfy this polynomial. Enjoy the puzzle of code-breaking your way through these math equations!