91Ó°ÊÓ

The elimination method is a popular technique for solving systems of linear equations. By manipulating the given equations, we aim to eliminate one of the variables, allowing for straightforward calculation. In our exercise, we have two equations:

• \(2x + 5y = -4\)
• \(3x - y = 11\)

First, we rearrange the second equation to solve for \(y\) in terms of \(x\):\[ y = 3x - 11 \]Next, we substitute this definition of \(y\) into the first equation, eliminating \(y\) entirely:\[ 2x + 5(3x - 11) = -4 \]Upon simplification, this yields a single equation with only \(x\):\[ 17x = 51 \]Solving for \(x\), we find \(x = 3\). With \(x\) known, we return to one of the original equations to find \(y\), resulting in \(y = -2\). Thus, the elimination method gives us a clear and reliable path to solving the system.

In systems of linear equations, a unique solution occurs when there is exactly one set of values for the variables that satisfies both equations simultaneously. For the given system:

• \(2x + 5y = -4\)
• \(3x - y = 11\)

Using the elimination method, we determined that \(x = 3\) and \(y = -2\) is the only solution that works for both equations.
This indicates a unique solution.
Unique solutions are common when two lines represented by equations intersect at a single point in a graph. Their slopes are different, ensuring they only cross once, unlike parallel lines or the same line, which would cause no or infinite solutions.

"No solution" in a system of equations implies that there is no point where both equations hold true simultaneously. This typically happens when the lines represented by the equations are parallel, meaning they never intersect. Parallel lines have the same slope but different y-intercepts; therefore, there is no shared solution. For this specific system, we verified a unique solution; hence, the lines intersect exactly once, indicating they are not parallel. However, should you face a similar system in the future revealing no solution; identify it by confirming that equations are scalar multiples of each other but differ in their constant terms.

Infinitely many solutions arise when two equations represent the same line, leading to every point on that line being a solution. This occurs when both equations are essentially the same, fundamentally scaled versions of one another. For example, multiplying both sides of an equation by a constant can create such a scenario. However, in our exercise, the elimination method and verification of solutions reveal that this is not the case, as we have a unique solution instead. Remember, a system with infinitely many solutions will have equations that are entirely consistent and collapse into each other after simplification, which is a distinct scenario from our current problem.