graphing method
Draw each equation as a picture on the same set of axes, and the solution reveals itself where the pictures cross. The graphing method treats each equation as a curve — for linear equations, a straight line — and reads off the system's solution as the point (or points) the graphs share.
For two lines, there are exactly three possibilities, and the graph makes them visible at a glance. The lines may cross at one point (one solution), run parallel and never meet (no solution), or lie perfectly on top of each other (infinitely many solutions). The geometry mirrors the algebra: a single crossing, a clean miss, or a complete overlap.
Graphing builds wonderful intuition and is the best way to see why a system behaves as it does. Its honest weakness is precision: by eye you may read an intersection as (2, 3) when the true point is (1.97, 3.04). For exact answers, confirm with substitution or elimination; use the graph to understand, and algebra to be sure.
Graphing y = x + 1 and y = −x + 5: the two lines cross at (2, 3), so that point is the solution. If instead the lines were y = x + 1 and y = x + 4, they would be parallel — no crossing, no solution.
Where the graphs meet is the answer.