4-40. Solve each of the following systems algebraically. What do the solutions tell you about each system? \( \begin{array}{l}\text { Visualizing the graphs may help with your description. } \\ \begin{array}{ll}\text { b. } y-7=-2 x\end{array} \\ \begin{array}{ll}4 x+2 y=14 & \text { d. } y=3(x-2)^{2}+3\end{array} \\ \\ \\ \\ 3(y-2 x-12\end{array} \)

To solve the given systems algebraically, we’ll start by rearranging each equation for easier visualization and analysis. For part b, the equation can be rewritten as \(y = -2x + 7\). Meanwhile, the second equation from d, \(y = 3(x-2)^{2} + 3\), represents a parabola that opens upwards with its vertex at (2, 3). If we solve these two equations together by setting them equal, we can substitute \(y\) from part b into part d: \(-2x + 7 = 3(x-2)^{2} + 3\). Simplifying this gives a quadratic equation that we can solve for \(x\). The solutions to these systems will likely reveal intersections or lack thereof, indicating whether the equations represent parallel lines, intersecting lines, or a combination. A visual graph would certainly help you understand these relationships further! Now, tackling the equation from part c, we want \(3(y - 2) = x - 12\), rewriting it to \(x = 3(y - 2) + 12\), and then following through to find corresponding y-values. This forms linear behavior, typically suggesting that it could intersect with previous equations significantly or reflect unique solutions based on the behavior of parabolas versus lines. Digging into solutions will not just show where these graphs intersect or diverge but also inform whether the system is consistent (one solution), inconsistent (no solution), or dependent (infinitely many solutions). So, sharpening those algebraic skills will really pay off!