\( 3 x+2 y=7 \) and \( -3 x+y=-1 \)

To solve the system of equations \( 3x + 2y = 7 \) and \( -3x + y = -1 \), we can use the method of substitution or elimination. By adding the two equations together, the \(3x\) and \(-3x\) terms cancel out. This will make it much easier to find the value of \(y\). We find that \(3x + 2y + (-3x + y) = 6\), simplifying to \(3y = 6\), which gives us \(y = 2\). Substituting \(y\) back into one of the original equations lets us solve for \(x\). Now, if you take \(y = 2\) and plug it into \(3x + 2(2) = 7\), you'll get \(3x + 4 = 7\), simplifying gives \(3x = 3\), so \(x = 1\). The solution to the system is \( (1, 2) \!). If you ever find yourself dealing with two equations, remember that checking your solutions by plugging them back into the original equations can confirm you've landed on the right answer. Avoid common mistakes like forgetting to simplify or misaligning your variables when you multiply or add; keeping a clear structure can save you from unnecessary confusion!