Solve the system of equations:
[tex]
\begin{array}{l}
y=2 x+1 \
-4 x+2 y=2
\end{array}
[/tex]
A. (2,3)
B. (-1, 1)
C. No solution
D. Infinite solutions
Answer (2)
Substitute the first equation into the second equation.
Simplify the equation and observe that it results in an identity ( 2 = 2 ).
Conclude that the system has infinite solutions since the equations are dependent.
The system has Infinite solutions .
Explanation
Analyze the problem We are given a system of two linear equations:
y = 2 x + 1 − 4 x + 2 y = 2
We need to determine if the system has a unique solution, no solution, or infinite solutions.
Solve the system Let's substitute the first equation into the second equation to solve for x :
− 4 x + 2 ( 2 x + 1 ) = 2
− 4 x + 4 x + 2 = 2
2 = 2
The equation simplifies to an identity, which means the system has infinite solutions.
Conclusion Since the substitution resulted in an identity ( 2 = 2 ), the system of equations has infinite solutions. This means that the two equations are dependent and represent the same line.
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, and modeling supply and demand in economics. For example, a company might use a system of equations to determine how many units of two different products they need to sell to reach a specific revenue target, given the prices of the products and the costs involved in producing them. Understanding how to solve systems of equations is crucial for making informed decisions in many fields.
The system of equations has infinite solutions as the two equations represent the same line. This is confirmed by substituting one equation into the other, leading to an identity. Therefore, the answer to the question is infinite solutions.
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