Math · College algebra · Worked example

Recognize a system with no solution

Solve the system x + 2y = 4 and 2x + 4y = 5.

x + 2 y = 4, 2 x + 4 y = 5

Try to eliminate x

Multiply the first equation by 2 to get 2x + 4y = 8, then subtract the second equation. Both variables disappear.

Subtracting the equations
	x	y	constant
2 × first equation	2	4	8
second equation	2	4	5
difference	0	0	3

Read the result

The difference says 0 = 3, which is false, so no pair (x, y) satisfies both equations: the system is inconsistent.

See why on the graph

Both lines have slope −1/2 but different y-intercepts, 2 and 5/4. Parallel lines never meet.

Result

There is no solution: the lines are parallel.

Your turn

How many solutions does the system 3x − y = 2 and 6x − 2y = 4 have?

Show the answer and explanation

Infinitely many.

The second equation is 2 times the first, so both describe the same line. Every point on 3x − y = 2, such as (0, −2) and (1, 1), solves the system.

\begin{array}{l} 3 (0) - (- 2) = 2 \\ 3 (1) - 1 = 2 \end{array}

Keep exploring

In Matrices & linear systems, change the 5 to 8. Now the second equation is twice the first, the studio reports infinitely many solutions, and they are x = 4 − 2t, y = t for any t.

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Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax College Algebra 2e — Systems of linear equations: two variables

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