- Represent the system of equations as: \(3x + 6y = 21\) and \(2x + 4y = 6\).
- Apply the elimination method by multiplying the first equation by 2 and the second by 3 to align the coefficients of \(x\).
- Observe the resulting equation \(0 = 24\), which is a contradiction.
- Conclude that there is no solution: \(\boxed{NO\ SOLUTION}\).
Explanation
- Analysis of the System
The system of equations provided is:
1) \(3x + 6y = 21\)
2) \(2x + 4y = 6\)
Our goal is to solve this system using the addition (elimination) method. We analyze the relationship between the coefficients of \(x\) and \(y\) in the two equations to see if they represent parallel, intersecting, or identical lines.
- Applying the Addition Method
To eliminate one of the variables, we multiply the equations by constants to make the coefficients of one variable opposites.
Multiply the first equation by 2:
\(2 \times (3x + 6y) = 2 \times 21 \implies 6x + 12y = 42\)
Multiply the second equation by 3:
\(3 \times (2x + 4y) = 3 \times 6 \implies 6x + 12y = 18\)
Now, subtract the second modified equation from the first:
\((6x + 12y) - (6x + 12y) = 42 - 18\)
\(0 = 24\)
- Conclusion
The result of our elimination is the equation \(0 = 24\). This is a mathematical contradiction, as zero can never equal 24. This indicates that there are no values of \(x\) and \(y\) that can satisfy both equations simultaneously. Geometrically, this means that the two equations represent parallel lines that never intersect. Therefore, there is no solution to this system.
Examples
Consider a scenario where you have two different paths defined by linear equations on a map. If path A is represented by the line \(3x + 6y = 21\) and path B is represented by the line \(2x + 4y = 6\), we can observe that these paths are parallel to each other. Because they have the same slope but different y-intercepts, they will never cross. Just as these paths never intersect, this system of equations has no point \((x, y)\) in common, meaning there is no solution to the system.