Solving Systems by Graphing

A system is two equations at once, and the solution is the point satisfying both — where the lines cross.

Graph both in slope-intercept form and read the intersection. Then check it in *both* equations.

Three outcomes:

• Lines cross once: one solution.
• Lines are parallel (same slope, different intercept): no solution.
• Lines are identical: infinitely many solutions.

Graphing is the best method for seeing what is going on, and the worst for accuracy. If the intersection is at (2.4, −1.7), you will not read it correctly off a hand-drawn grid. Use graphing to understand, algebra to solve.

A system of two linear equations in two variables corresponds to two lines, and the solution set is their intersection. The three cases — consistent and independent (one point), inconsistent (empty), consistent and dependent (the whole line) — are determined entirely by the slopes and intercepts.

Comparing y = m₁x + b₁ and y = m₂x + b₂: distinct slopes give exactly one solution regardless of intercepts; equal slopes with distinct intercepts give none; equal slopes and intercepts give infinitely many. In standard form, the system A₁x + B₁y = C₁ and A₂x + B₂y = C₂ is inconsistent when A₁/A₂ = B₁/B₂ ≠ C₁/C₂ and dependent when all three ratios agree.

Graphing is exact only for lattice-point intersections, and its real value is diagnostic: it makes the three cases visible and shows immediately whether an algebraic answer is plausible.

The same classification extends to larger systems and is formalised by the rank of the coefficient matrix in linear algebra.