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 \[\begin{array}{l}\;5x - 3y = 6\\\;8x - 12y = 3\end{array}\]

View Explanation

Log in for full answers

Similar Questions

Suppose in an LP, we have the following constraints: What is the x value at the intersection point between the two constraint boundary lines x+6 y>=8 and 7x + 5y <= 35 ? (Round your answer to 4 decimals. )

Suppose 𝐴 is a 3 × 4 matrix and the column vectors of 𝐴 span 𝑅 3 . Which of the following statements must be FALSE?

Types of Solutions for a System Solve the given system of equations. Choose the best answer. { 2 𝑥 + 𝑦 = 4 8 𝑥 + 4 𝑦 = 20

Solve the system Ax=b, where A=\left[\begin{array}{ccc} 3&4\\ {5}&{6}\end{array}\right] and b=\left[\begin{array}{ccc} 4.7\\ {7}\end{array}\right]. What value is the first entry of x?

Solve the following systems of equations. [math: 2x−3y=1] 2x-3y=1 [math: 3x+4y=−1] 3x+4y=-1

Solve the system of linear equations:

The linear equations: [math: 2x−7y=95x=−3]\begin{array}{l}\;2x - 7y = {\rm{ }}9\\\;5x\quad \quad \; = -3\end{array} can be written in matrix form as:

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler