Solving systems of equations by graphing

The graph of an equation is the set of points that satisfy it. A point that lies on both graphs satisfies both equations, so it is a solution of the system. To solve y = 2x − 1 and y = −x + 5, graph both lines and read off where they meet: (2, 3). Check: 2·2 − 1 = 3 and −2 + 5 = 3.

• One solution: lines with different slopes cross exactly once.
• No solution: parallel lines (same slope, different intercepts) never meet. The system is inconsistent.
• Infinitely many solutions: two equations describing the same line, such as y = 2x + 1 and 4x − 2y = −2, agree at every point.

The same method works for any curves. A line and a parabola can meet twice, once (tangent) or not at all; a circle and a parabola up to four times. The system x² + y² = 25, y = x² − 5 has three solutions: (0, −5) and (±3, 4). Algebra gets hard quickly here, but the graph shows every solution and the calculator refines each one.

By hand, reading coordinates off a grid is accurate to maybe a tenth of a unit, so it is best for counting solutions and estimating them. In the calculator, selecting a curve marks its intersections with every other curve, and each one is refined with Newton’s method to about twelve significant digits, so (2, 3) shows as exactly (2, 3), and irrational solutions are given to full precision.

For linear systems, substitution or elimination gives the exact answer. Substituting y = 2x − 1 into y = −x + 5 gives 2x − 1 = −x + 5, so x = 2 and y = 3. The Algebra tab can solve the resulting single equation for you.