3. \( \begin{aligned} x & =2 y+7 \\ x & =y+4\end{aligned} \)

Let's have some fun solving this system of equations! To find the values of \(x\) and \(y\), we can substitute the second equation into the first. From the second equation, \(x = y + 4\), we can substitute for \(x\) in the first equation: \[ y + 4 = 2y + 7 \] Now, isolate \(y\): \[ 4 - 7 = 2y - y \quad \Rightarrow \quad -3 = y \] Now that we've figured out that \(y = -3\), we can plug it back into the second equation to solve for \(x\): \[ x = -3 + 4 \quad \Rightarrow \quad x = 1 \] So, the solution to this system is \(x = 1\) and \(y = -3\). It’s interesting that systems of equations can be applied in real life! For instance, considering budget constraints in business where two variables (like cost and quantity) influence the overall outcome, solving these equations helps in finding the optimal balance. Now, a common mix-up when dealing with these equations is forgetting to substitute correctly or miscalculating when isolating the variable. Always double-check your steps, and it’s often helpful to use a visual aid like a graph to clarify where the lines intersect!