Question

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&2 x+y=5\\\&-6 x-3 y=-15\end{aligned} $$

Short answer

The system of linear equations has exactly one solution.

Step-by-step solution

Graph the First Line

The first line is represented by the equation \(2x + y = 5\). To graph it, it could be made into slope-intercept form, \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. By solving for \(y\), the equation becomes \(y = -2x + 5\), which indicates that the line crosses the y-axis at 5 and has a slope of -2.

Graph the Second Line

The second line is represented by the equation \(-6x - 3y = -15\). Again, rewriting this equation into the slope-intercept form results in \(y = 2x + 5\), with the line crossing the y-axis at 5 and having a slope of 2.

Analyze the Graph and Determine the Type of Solutions

Upon comparison, it is clear that the two equations have different slopes which means that the lines they represent will intersect at one point eventually. Therefore, the system of linear equations has exactly one solution.