Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent.
[tex]\begin{array}{l}
2 x+y=6 \\
x+\frac{1}{2} y=3
\end{array}[/tex]
The system has one solution.
The solution set is { }.
The system has no solution, { }.
The system is inconsistent.
The equations are dependent.
The system has infinitely many solutions.
The solution set is { [delimiter] x | x is any real number }.
The system is inconsistent.
The equations are dependent.
Answer (2)
Multiply the second equation by 2 and observe that it becomes identical to the first equation.
Recognize that the equations are dependent, indicating infinitely many solutions.
Express y in terms of x : y = 6 − 2 x .
State the solution set: {( x , 6 − 2 x ) ∣ x is any real number } .
Explanation
Analyzing the System of Equations We are given the following system of equations: 2 x + y = 6 x + f r a c 1 2 y = 3 We need to determine if the system has a unique solution, no solution (inconsistent), or infinitely many solutions (dependent).
Simplifying the Equations Let's multiply the second equation by 2: 2 ( x + f r a c 1 2 y ) = 2 ( 3 ) This simplifies to: 2 x + y = 6 Notice that this is the same as the first equation.
Determining the Nature of the Solutions Since both equations are identical, this means that the system has infinitely many solutions. The equations are dependent. We can express y in terms of x :
y = 6 − 2 x
Expressing the Solution Set Therefore, the solution set can be expressed as the set of all ( x , y ) such that y = 6 − 2 x , or as the set of all ( x , 6 − 2 x ) where x is any real number.
Final Answer The system has infinitely many solutions. The solution set is {( x , 6 − 2 x ) ∣ x is any real number } . The equations are dependent.
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, or modeling supply and demand in economics. In this case, the dependent system indicates that there are multiple combinations of quantities that satisfy the given conditions, providing flexibility in decision-making.
The given system of equations has infinitely many solutions because both equations are equivalent. The equations are dependent, and the solution set can be expressed as {(x, 6 - 2x) | x is any real number}.
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