To solve the given system of equations using the elimination method, we need to manipulate the equations in a way that allows us to eliminate one of the variables.
The given system of equations is:
[tex]\[
\begin{array}{rr}
-x + 4y = 3 \\
-2x + 8y = 6
\end{array}
\][/tex]
1. Identify the equations:
[tex]\[
\begin{align*}
\text{Equation 1:} & \quad -x + 4y = 3 \\
\text{Equation 2:} & \quad -2x + 8y = 6
\end{align*}
\][/tex]
2. Manipulate the equations to eliminate one variable:
Notice that Equation 2 is simply twice Equation 1:
[tex]\[
-2x + 8y = 2(-x + 4y) = 2 \times 3 = 6
\][/tex]
Since Equation 2 is a scalar multiple of Equation 1, the two equations are essentially the same line. This means they are dependent and represent the same set of points.
3. Analyze the systematic implications:
When two linear equations are dependent, they have infinitely many solutions that lie on the line represented by either equation.
Given the analysis above, we conclude:
B. There are infinitely many solutions.
