Which set of equations intersect at \( (2,-1) ? \) A. \( \left\{\begin{array}{l}y=3+x \\ y=-x+1\end{array}\right. \) B. \( \left\{\begin{array}{l}y=2 x-1 \\ y=1-x\end{array}\right. \) C. \( \left\{\begin{array}{l}y=3 x \\ y=2 x-1\end{array}\right. \) D. \( \left\{\begin{array}{l}y=2 x-5 \\ y=1-x\end{array}\right. \)

To figure out which set of equations intersects at the point \( (2, -1) \), we must substitute \( x = 2 \) into each equation and check if \( y \) equals -1. For option A: 1. \( y = 3 + 2 = 5 \) (not -1) 2. \( y = -2 + 1 = -1 \) (this one works) For option B: 1. \( y = 2(2) - 1 = 4 - 1 = 3 \) (not -1) 2. \( y = 1 - 2 = -1 \) (not a match) For option C: 1. \( y = 3(2) = 6 \) (not -1) 2. \( y = 2(2) - 1 = 4 - 1 = 3 \) (not a match) For option D: 1. \( y = 2(2) - 5 = 4 - 5 = -1 \) (this one works) 2. \( y = 1 - 2 = -1 \) (this also works) Thus, options A and D both intersect at \( (2, -1) \), but only D has both equations satisfying that point. So, the correct answer is **D**. A fun fact: finding intersections of equations is like connecting dots in a connect-the-dots puzzle; you never know what shape or picture you'll uncover until you join those lines! And if you're ever puzzled by these problems, remember that graphing can be your best friend—like plotting a treasure map to the hidden intersection!