To determine which of the given equations represents a line passing through the points [tex]\((4, -6)\)[/tex] and [tex]\((0, -4)\)[/tex], we need to check if both points satisfy each equation.
Let’s examine the given equations:
### Equation I: [tex]\( x + 2y = -6 \)[/tex]
1. Substituting [tex]\((4, -6)\)[/tex]:
[tex]\[
4 + 2(-6) = 4 - 12 = -8
\][/tex]
This simplifies to [tex]\(-8 \neq -6\)[/tex], so [tex]\((4, -6)\)[/tex] does not satisfy this equation.
2. Substituting [tex]\((0, -4)\)[/tex]:
[tex]\[
0 + 2(-4) = 0 - 8 = -8
\][/tex]
This simplifies to [tex]\(-8 \neq -6\)[/tex], so [tex]\((0, -4)\)[/tex] also does not satisfy this equation.
Since neither point satisfies Equation I, this equation does not represent the line passing through the points [tex]\((4, -6)\)[/tex] and [tex]\((0, -4)\)[/tex].
### Equation II: [tex]\( y - 1 = -\frac{1}{2}(x + 10) \)[/tex]
1. Substituting [tex]\((4, -6)\)[/tex]:
[tex]\[
-6 - 1 = -\frac{1}{2}(4 + 10)
\][/tex]
Simplifying the right side:
[tex]\[
-6 - 1 = -\frac{1}{2} \times 14 = -7
\][/tex]
This equation simplifies to [tex]\(-7 = -7\)[/tex], so [tex]\((4, -6)\)[/tex] satisfies Equation II.
2. Substituting [tex]\((0, -4)\)[/tex]:
[tex]\[
-4 - 1 = -\frac{1}{2}(0 + 10)
\][/tex]
Simplifying the right side:
[tex]\[
-4 - 1 = -\frac{1}{2} \times 10 = -5
\][/tex]
This equation simplifies to [tex]\(-5 = -5\)[/tex], so [tex]\((0, -4)\)[/tex] also satisfies Equation II.
Since both points satisfy Equation II, this equation does represent the line passing through the points [tex]\((4, -6)\)[/tex] and [tex]\((0, -4)\)[/tex].
### Conclusion:
The equation that represents the line passing through the points [tex]\((4, -6)\)[/tex] and [tex]\((0, -4)\)[/tex] is
[tex]\[ \text{II only} \][/tex]
