It looks like you provided a list of systems of linear equations, but the formatting and structure are unclear. Let's organize each system clearly and examine under what conditions each system has one solution. For a system of linear equations to have exactly one solution, the determinant of the coefficient matrix must be non-zero.

Let’s consider each system:

a.

4x + ay &= 2, \\ 5x + 3y &= 7 \end{aligned}$$ The coefficient matrix is: $$\begin{pmatrix} 4 & a \\ 5 & 3 \end{pmatrix}$$ The determinant is $$\text{Det} = 4(3) - 5(a) = 12 - 5a$$. For a unique solution, $$12 - 5a \neq 0$$. Thus, $$a \neq \frac{12}{5}$$. **b.** $$\begin{aligned} 4x + 16y &= 4, \\ ax + 10y &= 4 \end{aligned}$$ The coefficient matrix is: $$\begin{pmatrix} 4 & 16 \\ a & 10 \end{pmatrix}$$ The determinant is $$\text{Det} = 4(10) - 16(a) = 40 - 16a$$. For a unique solution, $$40 - 16a \neq 0$$. Thus, $$a \neq \frac{40}{16} = 2.5$$. **c.** $$\begin{aligned} 2x - 3ay &= 18, \\ 5x - 8y &= 18 \end{aligned}$$ The coefficient matrix is: $$\begin{pmatrix} 2 & -3a \\ 5 & -8 \end{pmatrix}$$ The determinant is $$\text{Det} = 2(-8) - 5(-3a) = -16 + 15a$$. For a unique solution, $$-16 + 15a \neq 0$$. Thus, $$a \neq \frac{16}{15}$$. **d.** $$\begin{aligned} 2ax - 2y &= -18, \\ 5y + 8x &= -18 \end{aligned}$$ The coefficient matrix is: $$\begin{pmatrix} 2a & -2 \\ 8 & 5 \end{pmatrix}$$ The determinant is $$\text{Det} = 2a(5) - 8(-2) = 10a + 16$$. For a unique solution, $$10a + 16 \neq 0$$. Thus, $$a \neq -\frac{16}{10} = -1.6$$. **e.** $$\begin{aligned} 2ax + 5y &= 3, \\ 3x + 4y &= 4 \end{aligned}$$ The coefficient matrix is: $$\begin{pmatrix} 2a & 5 \\ 3 & 4 \end{pmatrix}$$ The determinant is $$\text{Det} = 2a(4) - 5(3) = 8a - 15$$. For a unique solution, $$8a - 15 \neq 0$$. Thus, $$a \neq \frac{15}{8}$$. **f.** The system provided is unclear, and the question mark needs clarification. **Conclusion:** For each system: - **a.** The system has one solution for $$a \neq \frac{12}{5}$$. - **b.** The system has one solution for $$a \neq 2.5$$. - **c.** The system has one solution for $$a \neq \frac{16}{15}$$. - **d.** The system has one solution for $$a \neq -1.6$$. - **e.** The system has one solution for $$a \neq \frac{15}{8}$$. - **f.** Needs clarification. Would you like more details on any of these or do you have any questions? --- **Relative Questions:** 1. How do you calculate the determinant of a 2x2 matrix? 2. What conditions make a system of linear equations have no solution? 3. What does it mean when the determinant of a matrix is zero? 4. Can a system of linear equations have infinite solutions? Under what conditions? 5. How can you solve a system of linear equations using matrices? **Tip:** Remember, if the determinant of the coefficient matrix is zero, the system may have either no solutions or infinitely many solutions.