Solve Systems of Linear Equations: Parallel Lines and No Solutions

Published on August 9, 2026

Algebra

Determine the number of solutions for a system of linear equations by comparing coefficients and identifying parallel lines.

Question

To determine how many solutions the system of equations has, we can analyze the two given linear equations:

Step 1: Compare the coefficients of the variables.
We can look at the ratio of the coefficients of

and

for both equations.

Notice that if we multiply the first equation by

, we get:

Step 2: Compare the resulting equations.
Now we compare this modified first equation with the second equation:

The left-hand sides of both equations are identical (

), but the right-hand sides are different (

Step 3: Interpret the result.
Since

cannot be equal to both

and

at the same time, there are no values of

and

that satisfy both equations simultaneously. Geometrically, this means the two lines are parallel and will never intersect.

Therefore, the system has no solutions.

The correct answer is (B).