91Ó°ÊÓ

In the context of systems of linear equations, understanding the slopes of lines is crucial. Each linear equation can be represented as a line on a coordinate plane. The slope of a line provides information about its angle of inclination with respect to the x-axis. For a linear equation of the form \(ax + by = e\), the slope is calculated as \(-\frac{a}{b}\). This formula comes from rearranging the equation into the slope-intercept form, \(y = mx + c\), where \(m\) is the slope.

The concept of slope is key when analyzing systems of equations. By calculating the slopes \(-\frac{a}{b}\) for \(ax + by = e\) and \(-\frac{c}{d}\) for \(cx + dy = f\), we can determine if the lines are parallel, identical, or intersecting at a unique point.

A system of linear equations will have a unique solution when the lines represented by these equations intersect at a single point. This point is where both equations are simultaneously satisfied, and no other points will do the same.

To achieve this unique intersection point, the key is that the lines should be non-parallel. Non-parallel lines are those that have different slopes, ensuring they eventually cross each other somewhere on the graph. In mathematical terms, for our system of equations, the inequality \(-\frac{a}{b} eq -\frac{c}{d}\) or equivalently, \(ad eq bc\), must hold true.

If this condition is met, you can be confident that the system has one and only one solution, as each line will take a different path through the graph.

The conditions for a system of linear equations to be solvable, specifically to have a unique solution, revolve around the relationship between their coefficients. For the given system:

• If \(ad eq bc\), the lines intersect at exactly one point, leading to a unique solution.
• If \(ad = bc\), the situation changes. The lines could be parallel with no points in common, indicating no solutions.
• Alternatively, if the entire equations are multiples of each other, the lines will overlap perfectly and result in infinitely many solutions.

Understanding this relationship between the coefficients helps in determining whether the set of equations will provide a feasible and unique solution, one that covers specific situations of overlap or parallelism.