Problem 48 - Algebra Form and Function

Question

For the system $$ \left\\{\begin{array}{l} 2 x+3 y=5 \\ 4 x+6 y=n \end{array}\right. $$ what must be true about \(n\) in order for there to be many solutions?

Step-by-Step Solution

Answer

Answer: The system of linear equations will have many solutions if the parameter n=10.

Step 1: Analyze the system

We have the given system of linear equations: $$ \left\\{\begin{array}{l} 2 x+3 y=5 \\\ 4 x+6 y=n \end{array}\right. $$

Step 2: Express one of the equations in terms of the other

We can manipulate equation 1 to help us determine the parameter n. Let's multiply equation 1 by 2 to match the coefficients of x and y in equation 2. $$ 2(2x + 3y) = 2(5) \Rightarrow 4x + 6y = 10 $$

Step 3: Determine the conditions for n

Now, we compare this result with the second equation in the original system to find the conditions for n that will make the system consistent and dependent (many solutions). If the system is consistent and dependent, then both equations represent the same line. The second equation in the original system is: $$ 4x + 6y = n $$ Comparing with the result obtained in Step 2, we can see that they are multiples of each other when: $$ n = 10 $$

Step 4: Conclusion

The system of linear equations will have many solutions if the parameter n satisfies: $$ n = 10 $$

Key Concepts

Dependent Systems

In the context of systems of linear equations, a dependent system is one where the equations describe the same geometric figure, typically a line. As a result, there are infinitely many solutions to the system since each point on the line is a solution.
When you analyze a system like \[ \begin{cases} 2x + 3y = 5 \ 4x + 6y = n \end{cases} \] you’ll notice that the second equation is a multiple of the first.
This indicates they are dependent. To confirm it, both must have the same proportional constant when comparing their corresponding coefficients.

To illustrate, multiplying the entire first equation by 2 gives:\[ 4x + 6y = 10 \] Compare this with the second equation to determine the value of \( n \) that makes the system dependent. Here, \( n \) must equal 10 for the equations to be equivalent. This confirms that at \( n = 10 \), the lines coincide, leading to a dependent system with infinitely many solutions.

Consistent Systems

A consistent system of equations is one in which there is at least one set of values for the variables that satisfies all the equations simultaneously. When both equations in a system are different representations of the same line, the system is not only consistent, but it is also dependent.
For example, in the system:\[ \begin{cases} 2x + 3y = 5 \ 4x + 6y = n \end{cases} \] we saw that by ensuring \( n = 10 \), both equations describe the very same line.

To guarantee consistency, you should check if multiplying or manipulating the coefficients of one equation results in the other. If so, the system is consistent. Therefore, for these equations, making \( n = 10 \) ensures that not only does a solution exist, but there are infinitely many (each point on the line).
This overlap confirms they are consistent, emphasizing reliability and solvability in the system of equations.

Parameters in Equations

Parameters in systems of linear equations are values that can alter the system's properties and its set of solutions. These are constants like \( n \) in our problem that, when modified, impact the consistency and dependency of the system.
Take the system: \[ \begin{cases} 2x + 3y = 5 \ 4x + 6y = n \end{cases} \] By treating \( n \) as a parameter, you assess what value makes the system consistent and dependent, meaning the lines overlap entirely.

Applying algebraic manipulations, such as multiplying the first equation by 2, we equate it with the second equation to find \[ n = 10 \].
When the parameter \( n \) satisfies this condition, the system reflects infinitely many solutions, all lying on the same line. Understanding the role of parameters helps in identifying not just when solutions exist, but in determining the exact conditions needed for specific solution types.