To determine the nature of the graph of the given system of equations, we need to analyze the relationships between the coefficients of the equations. Here are the given equations:
1. [tex]\(-2x + y = 3\)[/tex]
2. [tex]\(4x + 2y = 2\)[/tex]
We can rewrite these equations in standard form [tex]\(Ax + By = C\)[/tex]:
1. [tex]\(-2x + y = 3\)[/tex]
2. [tex]\(4x + 2y = 2\)[/tex]
### Step 1: Calculate the Determinant
The determinant helps us determine if the lines intersect, are parallel, or overlap. The coefficient matrix for the given system of equations is:
[tex]\[
\begin{pmatrix}
-2 & 1 \\
4 & 2
\end{pmatrix}
\][/tex]
The determinant [tex]\(\Delta\)[/tex] of this matrix is calculated as:
[tex]\[
\Delta = a1 \cdot b2 - a2 \cdot b1 = (-2) \cdot 2 - 4 \cdot 1 = -4 - 4 = -8
\][/tex]
Since the determinant is not zero ([tex]\(\Delta = -8\)[/tex]), it indicates that the system of equations represents lines that intersect at a single point. Lines with a non-zero determinant are neither parallel nor overlapping; they must intersect.
### Conclusion
Based on the calculations, the graph of the system of equations given by:
[tex]\[
\begin{array}{l}
-2x + y = 3 \\
4x + 2y = 2
\end{array}
\][/tex]
represents Intersecting lines.
