Systems of Two Linear Equations – Solving Graphically

What it shows

Each linear equation ax + by = c in two unknowns has a straight line as its graph, and a solution of the system is a point lying on both lines at once. Two lines in a plane can intersect once, be parallel or coincide, so a system of two linear equations has exactly one solution, no solution or infinitely many. The elimination method gives the same answer algebraically: combining multiples of the equations leads to a single value of x, to 0 = k with k ≠ 0, or to 0 = 0. Nearly parallel lines meet far from the origin, which gives solutions with large values.

How to use

Start with the example buttons Intersecting, Parallel and Coincident and compare each picture with the stated number of solutions. Then drag the round points on each line, or change a₁, b₁, c₁ and a₂, b₂, c₂, to build your own system. Use Show/hide steps to follow the elimination method and check that the intersection point M satisfies both equations. Finally, make the lines almost parallel and watch M move away.

Parameters you can change

• a₁ (equation 1) -10–10
• b₁ (equation 1) -10–10
• c₁ (equation 1) -10–10
• a₂ (equation 2) -10–10
• b₂ (equation 2) -10–10
• c₂ (equation 2) -10–10

Questions to explore

1. Does multiplying an equation by a nonzero number change its line, and when do two equations give the same line?
2. Which result in the elimination steps tells you that the system has no solution?
3. How can you check that a solution read from the graph is exact?