For what value of a will the system have of equations below have one solution:
4 1,2,
5 3 7;
x ay
x y
 + = 
 + =
a. b.
4 16,
4 10;
ax y
x ay
 + = 
 + =
c.
2 3,
5 8 18;
x ay
x y
 − =

 − =
d.
2 2,
5 8 18;
ax y
y x
 − = −

 + =
e.
2 5,
3 4;
ax y
x y
 + = 
 + =
f.
3 4 8,
11 6 ?

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.**
\[
\begin{aligned}
4x + ay &= 2, \\
5x + 3y &= 7
\end{aligned}
\]
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.

Explore the conditions under which systems of linear equations have exactly one solution. Learn about determinants of 2x2 matrices and their role in determining uniqueness of solutions. Suitable for high school students studying linear algebra.

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