Number of Solutions: Consistent, Dependent and Inconsistent Systems

A system of linear equations is consistent if it has at least one solution and inconsistent if it has none, and a consistent system is dependent if it has infinitely many solutions. In brief: two lines with different slopes give exactly one solution, parallel lines with different intercepts give none, and two equations describing the same line give infinitely many. For example, x + y = 3 and 2x + 2y = 6 describe one line, so every point of it is a solution.

Before solving a system it helps to know what to expect. The number of solutions can be read from the coefficients without doing the full calculation, and it can be detected in elimination when a variable disappears. This guide explains the three cases, how to recognise them with a comparison of coefficients and how to write the solution set.

What are the three possible cases?

For two linear equations in two unknowns there are exactly three cases. Geometrically each equation is a line, and two lines can intersect, be parallel or coincide.

• Exactly one solution: the lines intersect. The system is consistent and independent.
• No solution: the lines are parallel and distinct. The system is inconsistent.
• Infinitely many solutions: the lines coincide. The system is consistent and dependent.

A system of linear equations can never have exactly two or exactly three solutions. If two different points satisfy both equations, the whole line through them does.

How can you recognise the case from the coefficients?

For the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, with all coefficients nonzero, compare the ratios:

• If a₁/a₂ ≠ b₁/b₂: one solution.
• If a₁/a₂ = b₁/b₂ ≠ c₁/c₂: no solution.
• If a₁/a₂ = b₁/b₂ = c₁/c₂: infinitely many solutions.

Example 1

2x + 3y = 5 and 4x + 6y = 7. The ratios are 2/4 = 1/2 and 3/6 = 1/2, but 5/7 ≠ 1/2. The lines are parallel, so there is no solution. Check: doubling the first equation gives 4x + 6y = 10, which contradicts 4x + 6y = 7.

Example 2

x − 2y = 4 and 3x − 6y = 12. The ratios are 1/3, −2/−6 = 1/3 and 4/12 = 1/3. All are equal, so the equations describe the same line.

Example 3

3x + y = 5 and x − 2y = 4. Here 3/1 ≠ 1/(−2), so there is exactly one solution. Elimination gives x = 2, y = −1.

What does each case look like in elimination?

When you apply elimination or substitution, a variable can disappear along with the other:

• If you get a false statement, such as 0 = 6, the system has no solution.
• If you get a true statement, such as 0 = 0, there are infinitely many solutions.
• If you get an equation like 5x = 10, there is exactly one solution.

For 2x + 3y = 5 and 4x + 6y = 7, multiply the first by −2 and add: (−4x − 6y) + (4x + 6y) = −10 + 7, which gives 0 = −3. False, so no solution. For x − 2y = 4 and 3x − 6y = 12, multiply the first by −3 and add: 0 = 0. True, so infinitely many solutions.

How do you describe infinitely many solutions?

Solve one equation for one variable and use a free parameter. For x − 2y = 4, write x = 4 + 2y, with y any real number. Or take y = t and write the solutions as (4 + 2t, t). For t = 0 this gives (4, 0), for t = 1 it gives (6, 1) and for t = −1 it gives (2, −1). Each of these satisfies 3x − 6y = 12 as well.

Summary table of the three cases

Where does the classification come from?

The three cases were understood in connection with the geometric picture of lines in the plane. The general theory for any number of equations was developed in the nineteenth century using determinants and later matrices, with the rank of a matrix giving a complete criterion.

Where does it matter in practice?

An inconsistent system in a model is a warning: the conditions contradict each other, for example two price relations that cannot both be true. A dependent system means that a condition is redundant and the problem is under-determined, so more information is needed. Engineers and scientists check consistency before relying on a calculated solution.

Which mistakes are common?

Thinking 0 = 0 means no solution

It means every value works. The true statement indicates infinitely many solutions.

Comparing only slopes

Equal slopes mean parallel or identical lines. Compare the intercepts or the constant terms to decide.

Giving a single pair for a dependent system

Write the whole solution set using a parameter.

Dividing by zero in the ratio test

If a coefficient is 0, do not use the ratios. Solve by elimination instead.