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Given the linear system:

[tex]\[
\begin{aligned}
y & = -6x + 2 \\
-12x - 2y & = -4
\end{aligned}
\][/tex]

How many solutions does this linear system have?

A. one solution: [tex]\((0,0)\)[/tex]

B. one solution: [tex]\((1,-4)\)[/tex]

C. no solution

D. infinite number of solutions



Answer :

GinnyAnswer
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

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