Examplo \#1-Solvo Linear-Quadratio Systems of Equations Using Substitution: What aro the solutions to the systom of equations below? \[ \begin{array}{l}y=x^{2}-6 x+9 \\ y+x=5\end{array} \]
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To tackle this system of equations, we start by noticing that the first equation, \( y = x^2 - 6x + 9 \), can be simplified: it is actually a perfect square! It rewrites to \( y = (x - 3)^2 \). Now, for the second equation \( y + x = 5 \), we can solve for \( y \): \( y = 5 - x \). By equating both expressions for \( y \), we get \( (x - 3)^2 = 5 - x \). This is where the fun begins, as we can solve for \( x \) and ultimately find the solutions. After setting up the equation \( (x - 3)^2 = 5 - x \) and rearranging it into standard quadratic form, we can solve for \( x \) using the quadratic formula or by factoring. This will lead us to the potential \( x \) values, from which we can then derive the corresponding \( y \) values to complete the solution set for the system of equations! Once solved, you'll find the solutions are \( (1, 4) \) and \( (5, 0) \). So, keep your pencils sharp and have fun solving!
