System of equations

A system of two linear equations has exactly one solution when the two lines cross, no solution when they are parallel, and infinitely many when the two equations describe the same line. You can solve it by substitution, by elimination, or by writing both as y = ... and setting them equal. All three give the same answer, because all three describe the same crossing point.

Three methods, one crossing point

Substitution, elimination and comparison feel like three unrelated tricks from three different lessons. All three hunt for the same point: the place where both equations are true at once.

Switch method in the tool and the graph does not move. Only the working underneath changes. That is the point: pick the method that suits the numbers. A coefficient of 1 in front of x makes substitution quickest. Two x terms that nearly cancel already make elimination quickest.

No solution is not a mistake

Set b₂ so the second line runs parallel to the first. The two lines now travel side by side and never meet, so the system has no solution.

In your working this shows up as an impossible line such as 0 = 5. That is not an arithmetic slip, it is the answer. Now slide c₂ until the lines lie on top of each other and you get 0 = 0 instead: every point on the line works, so there are infinitely many solutions.

The determinant tells you in advance

At the bottom you can see a₁b₂ - a₂b₁. As long as that number is not zero, the lines cross and there is exactly one solution.

Once it hits zero the lines are parallel, and you only have to work out which of the two cases you have. It makes a quick check: compute the determinant first and you know straight away whether it is worth going further.

Frequently asked questions

What is the substitution method?

You isolate one unknown in one equation, for instance x = (c - by) / a, and put that expression into the other equation. What is left is a single equation in a single unknown.

When should you use elimination?

When the coefficients of x or y are easy to make opposite. You scale both equations so that one unknown disappears when you add or subtract them. With larger numbers this is usually tidier than substitution.

How is the comparison method different?

You write both equations as y = ... and set the right-hand sides equal to each other. It works well when both equations are already close to that form.

How do you know a system has no solution?

When the working leads to something impossible, such as 0 = 5. On the graph that is two parallel lines. If you end up with 0 = 0 instead, the lines coincide and there are infinitely many solutions.

Can you solve a system graphically?

Yes, and that is what this tool shows. Draw both lines and read off the crossing point. That is fine for whole-number answers, but you will never read a fraction exactly off a graph. Check those with one of the three methods.

More interactive maths tools

• Slope of a line →Drag two points and see Δy/Δx as a staircase, with the equation and the perpendicular alongside.
• Discriminant and parabola →Slide a, b and c and see D decide whether the parabola cuts, touches or misses the axis.
• Unit circle →Drag the angle and see sin, cos and tan as lengths, with the exact values.