Which of the following systems of linear equations does not have a unique solution? I $$\begin{array}{l}5x - 5y = 6\2x + 6y = 3\end{array}$$ II $$\begin{array}{l}\;23x + 23y = 6\\;25x + 25y = 3\end{array}$$ III $$\begin{array}{l}\;5x - 3y = 6\\;8x - 12y = 3\end{array}$$

Question

Which of the following systems of linear equations does not have a unique solution? I  $$\begin{array}{l}5x - 5y = 6\2x + 6y = 3\end{array}$$ II $$\begin{array}{l}\;23x + 23y = 6\\;25x + 25y = 3\end{array}$$ III $$\begin{array}{l}\;5x - 3y = 6\\;8x - 12y = 3\end{array}$$

BlackTom AI · Accepted Answer

We need to examine each labeled system I, II, and III to determine whether it has a unique solution, infinitely many solutions, or no solution at all.

Option I: 5x - 5y = 6 and 2x + 6y = 3.
- The first equation simplifies to x - y = 6/5. The second equation is 2x + 6y = 3, which can be rewritten as x + 3y = 3/2 after dividing by 2. Solving these two linear equations in two variables yields a single intersection point, so there is a unique solution. The lines are not parallel and do intersect at one point.

Option II: 23x + 23y = 6 and 25x + 25y = 3.
- These equations can be simplified by dividing by their coefficients: x + y = 6/23 and x + y = 3/25. The left-hand sides are identical after simplification, which means the two lines are parallel (they have the same slope) but are shifted to different intercepts unless the right-hand sides were equal. Here 6/23 ≠ 3/25, so the two equations express contradictory conditions for x and y. There is no pair (x, y) that satisfies both equations simultaneously, so this system has no solution.

Option III: 5x - 3y = 6 and 8x - 12y = 3.
- Solve the first equation for one variable, for instance x = (6 + 3y)/5, and substitute into the second: 8[(6 + 3y)/5] - 12y = 3. Clearing denominators gives 48 + 24y - 60y = 15, which simplifies to 48 - 36y = 15, leading to y = 11/12. Substituting back yields x = 7/4. Since we found a consistent solution for both equations, the system intersects at exactly one point, indicating a unique solution.

Summary of reasoning:
- In Option I, the two lines cross at a single point, giving a unique solution.
- In Option II, the two lines are parallel and inconsistent with each other, yielding no solution.
- In Option III, the two lines intersect at a single point as well, giving a unique solution.

Therefore, among the three systems, only Option II fails to have a unique solution (it has no solution).

Which of the following systems of linear equations does not have a unique solution? I \[\begin{array}{l}5x - 5y = 6\\2x + 6y = 3\end{array}\] II \[\begin{array}{l}\;23x + 23y = 6\\\;25x + 25y = 3\end{array}\] III \[\begin{array}{l}\;5x - 3y = 6\\\;8x - 12y = 3\end{array}\]

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