To determine the nature of the solutions for the system of equations:
[tex]\[
\begin{cases}
2x - y = 1 \\
4x - 2y = 2
\end{cases}
\][/tex]
we need to perform the following steps:
1. Write the coefficients of the equations:
[tex]\[
\begin{array}{ccc}
2x & -y &= 1 \\
4x & -2y &= 2
\end{array}
\][/tex]
Here, the coefficients [tex]\(a_1, b_1, a_2, b_2\)[/tex] are:
[tex]\[
a_1 = 2, \quad b_1 = -1, \quad a_2 = 4, \quad b_2 = -2
\][/tex]
2. Calculate the determinant of the coefficient matrix:
[tex]\[
\text{Determinant} = a_1 \cdot b_2 - a_2 \cdot b_1
\][/tex]
Substituting the given values:
[tex]\[
\text{Determinant} = 2 \cdot (-2) - 4 \cdot (-1) = -4 + 4 = 0
\][/tex]
3. Analyze the determinant:
- If the determinant is zero, this indicates that the lines represented by these equations are either parallel or coincident.
4. Check for consistency:
If the determinant is zero, the system may have either no solution or infinitely many solutions. To distinguish between these cases, we examine the ratios of the coefficients and the constants.
Consider the ratios:
[tex]\[
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
\][/tex]
Substituting the values:
[tex]\[
\frac{2}{4} = \frac{-1}{-2} = \frac{1}{2}, \quad \text{or} \quad \frac{1}{2} = \frac{1}{2}= \frac{1}{2}
\][/tex]
All three ratios are equal, meaning:
[tex]\[
\frac{2}{4} = \frac{-1}{-2} = \frac{1}{2}
\][/tex]
Since the ratios of the coefficients [tex]\(a_1/a_2\)[/tex], [tex]\(b_1/b_2\)[/tex], and constants [tex]\(c_1/c_2\)[/tex] are equal, it indicates that the system of equations is consistent and has infinitely many solutions.
Therefore, the statement that best describes the solution for this system of equations is:
[tex]\[
\text{The system has infinitely many solutions.}
\][/tex]
