IB Mathematics Analysis and Approaches Question Type 4: for Given Parameters in the Linear Equations, Find the Value of the Parameter That Will Result in No Solution, One Solution or Infinite Number of Solutions Exercises

Classify the solutions in terms of $a$ and $b$ for the system

$$\{\begin{array}{l}ax+y=2\\ 2x+ay=b\end{array}$$

Determine all $a$ for which the system has no solution, exactly one solution, or infinitely many solutions:

$$\{\begin{array}{l}ax+y+z=1\\ x+ay+z=1\\ x+y+az=1\end{array}$$

Find all $a$ and $b$ for which the $3\times 3$ system has no solution, exactly one solution, or infinitely many solutions: $$\{\begin{array}{l}x+y+z=1\\ ax+y+z=1\\ x+ay+z=b\end{array}$$

Find all values of the parameters $a$ and $k$ for which the system has no solution, exactly one solution, or infinitely many solutions:

$$\{\begin{array}{l}ax+2y=3\\ 2x+4y=k\end{array}$$

Determine for which parameters the system has no solution, exactly one solution, or infinitely many solutions:

$$\{\begin{array}{l}x+y+z=1\\ 2x+ay+z=2\\ 3x+2y+bz=c\end{array}$$

Find all $a$ and $k$ yielding no solution, one solution, or infinitely many solutions:

$$\{\begin{array}{l}(2-a)x+y=3\\ 4x+(2-a)y=k\end{array}$$

Find all $k$ and $b$ so that the system has no solution, exactly one solution, or infinitely many solutions:

$$\{\begin{array}{l}(1-k)x+y=2\\ 2x+(2-2k)y=b\end{array}$$

For which values of $a$ and $b$ does the system have no solution, exactly one solution, or infinitely many solutions?

$$\{\begin{array}{l}(a-1)x+y=2\\ 2x+(b-2)y=3\end{array}$$

Classify the solutions in terms of $a$ and $b$ for the system

$$\{\begin{array}{l}(a+1)x+2y=5\\ 2(a+1)x+(4+b)y=10\end{array}$$

Determine all $a$ and $b$ for which the system has no solution, exactly one solution, or infinitely many solutions:

$$\{\begin{array}{l}(2a)x+3y=1\\ (4a)x+by=2\end{array}$$

Classify the solutions.