∆= 𝑎11𝑎22 − 𝑎12𝑎21.
a. 5𝑥 − 4𝑦 = 19	b. 4𝑥 − 6𝑦 = 6
−2𝑥 + 3𝑦 = −9	6𝑥 − 3𝑦 = 5


Let's break down the two systems of linear equations given and compute the determinant (∆) of each.

**System (a):**
1. \( 5x - 4y = 19 \)
2. \( 4x - 6y = 6 \)

To represent this system in matrix form:
\[
\begin{pmatrix}
5 & -4 \\
4 & -6
\end{pmatrix}
\]
The determinant (∆) for this system is:
\[
\Delta = a_{11}a_{22} - a_{12}a_{21}
\]
Where:
- \( a_{11} = 5 \)
- \( a_{12} = -4 \)
- \( a_{21} = 4 \)
- \( a_{22} = -6 \)

So,
\[
\Delta = (5)(-6) - (-4)(4) = -30 + 16 = -14
\]

**System (b):**
1. \( -2x + 3y = -9 \)
2. \( 6x - 3y = 5 \)

In matrix form:
\[
\begin{pmatrix}
-2 & 3 \\
6 & -3
\end{pmatrix}
\]
The determinant (∆) for this system is:
\[
\Delta = (-2)(-3) - (3)(6) = 6 - 18 = -12
\]

### Summary of the Determinants:
- **System (a):** \( \Delta = -14 \)
- **System (b):** \( \Delta = -12 \)

Would you like further details or clarification on any step?

### Related Questions:
1. How do we solve systems of equations using determinants (Cramer's rule)?
2. How can you find the inverse of a 2x2 matrix using its determinant?
3. What does a zero determinant tell us about the system of equations?
4. How do you solve a system of linear equations using matrix methods?
5. What is the geometric interpretation of a determinant in 2D?

### Tip:
When calculating a determinant, always ensure the matrix is square (same number of rows and columns), as determinants are only defined for square matrices.

∆= 𝑎11𝑎22 − 𝑎12𝑎21. a. 5𝑥 − 4𝑦 = 19 b. 4𝑥 − 6𝑦 = 6 −2𝑥 + 3𝑦 = −9 6𝑥 − 3𝑦 = 5

Learn how to calculate the determinants of 2x2 matrices for systems of linear equations. This example walks through solving two systems, calculating ∆ = a11 * a22 - a12 * a21, and explains key concepts like determinants and Cramer's Rule. Suitable for students in Grades 9-12.

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