Solve by Substitution.
$\left\{\begin{array}{ll}
x & =-2 y+9 \\
3 x+6 y & =27
\end{array}\right.$
A. One solution
B. No solution
C. Infinite number of solutions
Answer (1)
The system of equations has infinitely many solutions because substituting the first equation into the second results in an identity (27 = 27). This means the two equations are dependent and represent the same line. Therefore, there are infinitely many pairs of (x, y) that satisfy both equations.
Infinite number of solutions
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y . Our goal is to solve this system using the substitution method and determine whether there is one solution, no solution, or an infinite number of solutions. The given equations are:
Equation 1: x = − 2 y + 9 Equation 2: 3 x + 6 y = 27
Substitution We will substitute the expression for x from Equation 1 into Equation 2. This means replacing x in Equation 2 with ( − 2 y + 9 ) .
3 x + 6 y = 27 3 ( − 2 y + 9 ) + 6 y = 27
Simplify the equation Now, we simplify the equation:
− 6 y + 27 + 6 y = 27 27 = 27
Determine the nature of the solution The equation simplifies to 27 = 27 , which is an identity. This means that the equation is always true, regardless of the values of x and y . This indicates that the system has infinitely many solutions.
Express the solution set Since we have infinitely many solutions, we can express the solution set in terms of one variable. From Equation 1, we have x = − 2 y + 9 . This equation represents the relationship between x and y for all solutions to the system.
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, understanding that there are infinite solutions means there are many possible combinations of two items that satisfy a given set of conditions, which can be useful in resource allocation or decision-making processes.
