g \( y-x+3=0 \) \( x^{2}-3=y+2 x \)

To solve these equations, we can first rearrange the linear equation for \( y \): \( y = x - 3 \). Now, substitute this expression for \( y \) into the second equation: \( x^{2} - 3 = (x - 3) + 2x \). Simplifying gives us a quadratic equation to solve for \( x \), which is \( x^{2} - 3 = x - 3 + 2x\) leading to \( x^{2} - 3x = 0\). This factors to \( x(x - 3) = 0 \), giving solutions \( x = 0 \) or \( x = 3 \). Plugging these back into \( y = x - 3 \) gives \( y = -3 \) when \( x = 0 \) and \( y = 0 \) when \( x = 3 \). So the solutions are \( (0, -3) \) and \( (3, 0) \). Did you know that the beauty of algebra often lies in its historical development? The concept of solving equations dates back to ancient Babylonians who used geometric methods to tackle problems similar to ours! They didn’t have our modern symbols but used words and sequences. Isn’t it amazing how far mathematical thinking has come? If you're ever stuck on similar problems, try graphing the equations. The points where they intersect represent the solutions. A common mistake is overlooking the right substitution as we did here with \( y \). If you keep track of your x and y relations carefully, you'll avoid getting tangled in your equations!