QuestionWhich statement best describes the solution of the system of equations shown?
The system has no solutions.
The system has intinitely many solutions.
The system has exactly two solutions.
The system has exactly one solution.
Studdy Solution
STEP 1
1. The system of equations is given by:
$ \begin{array}{c}
2x - y = 1 \\
4x - 2y = 2
\end{array}
\]
2. The goal is to determine the nature of the solution to the system of equations: no solutions, infinitely many solutions, exactly two solutions, or exactly one solution.
3. We will use algebraic methods to analyze the system, such as substitution, elimination, or matrix methods.
STEP 2
1. Analyze the given system of equations to determine if they are consistent and dependent or inconsistent.
2. Use algebraic manipulation to simplify the system and determine the relationship between the two equations.
3. Conclude the nature of the solutions based on the simplified equations.
STEP 3
Rewrite the given system of equations for clarity:
STEP 4
Observe that Equation 2 is a multiple of Equation 1. Specifically, multiply Equation 1 by 2: This simplifies to:
STEP 5
Compare the simplified Equation 2 with the original Equation 2: Both equations are identical, meaning they are dependent.
STEP 6
Since the two equations are dependent, this implies that they represent the same line. Therefore, there are infinitely many solutions that lie on this line.
STEP 7
Conclude that the system of equations has infinitely many solutions.
Solution:
The system of equations has infinitely many solutions.
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