To determine the nature of the relationship between the graph of the given system of equations, we start by analyzing and comparing the two equations:
1. [tex]\(-2x + y = 3\)[/tex]
2. [tex]\(4x + 2y = 2\)[/tex]
First, let's simplify the second equation by dividing every term by 2:
[tex]\[ 4x + 2y = 2 \implies 2x + y = 1\][/tex]
Now we have the simplified system of equations:
1. [tex]\(-2x + y = 3\)[/tex]
2. [tex]\(2x + y = 1\)[/tex]
Next, let’s use the method of elimination to find whether these equations have a common solution (i.e., whether the lines intersect).
First, add the two equations:
[tex]\[
(-2x + y) + (2x + y) = 3 + 1
\][/tex]
This simplifies to:
[tex]\[
0x + 2y = 4 \implies 2y = 4 \implies y = 2
\][/tex]
Now substitute [tex]\(y = 2\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. Let’s use the second simplified equation [tex]\(2x + y = 1\)[/tex]:
[tex]\[
2x + 2 = 1 \implies 2x = -1 \implies x = -\frac{1}{2}
\][/tex]
So, we find that the solution to the system is:
[tex]\[ x = -\frac{1}{2}, \quad y = 2 \][/tex]
Since we found a unique solution [tex]\((- \frac{1}{2}, 2)\)[/tex], the lines intersect at this point. Therefore, the graph of this system of equations represents two intersecting lines.
Hence, the correct answer is:
Intersecting lines
