To determine how many solutions the given system of linear equations has, we will solve it step-by-step. The system is:
[tex]\[
\begin{aligned}
y &= -6x + 2 \quad \text{(Equation 1)} \\
-12x - 2y &= -4 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
### Step 1: Substitute Equation 1 into Equation 2
First, we'll substitute the expression for [tex]\(y\)[/tex] from Equation 1 into Equation 2.
[tex]\[
y = -6x + 2
\][/tex]
Substitute [tex]\(y\)[/tex] into Equation 2:
[tex]\[
-12x - 2(-6x + 2) = -4
\][/tex]
### Step 2: Simplify the resulting equation
Let's simplify the left-hand side of the substituted equation:
[tex]\[
-12x - 2(-6x + 2) = -12x + 12x - 4
\][/tex]
Combine like terms:
[tex]\[
-12x + 12x - 4 = -4
\][/tex]
This simplifies to:
[tex]\[
-4 = -4
\][/tex]
### Step 3: Analyze the result
The simplified equation [tex]\(-4 = -4\)[/tex] is a tautology (always true), meaning that the two original equations are dependent. This implies that any [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy Equation 1 will also satisfy Equation 2. Hence, there are infinitely many solutions.
### Conclusion
Therefore, the system of equations has an infinite number of solutions. The correct answer is:
- infinite number of solutions
