Question

Graph the system of linear equations. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&2 x+y=7\\\&3 x-y=-2\end{aligned} $$

Short answer

The system of linear equations has exactly one solution, which is represented by the point of intersection on the graph.

Step-by-step solution

Rearrange the equations in y = mx + c form

Rearrange both equations into slope-intercept form, that is, y = mx + c. For the first equation: \( y = -2x + 7 \) For the second equation: \( y = 3x + 2 \)

Graph the equations

Now, plot these two equations on the same graph. The line \( y = -2x + 7 \) will slope downwards and cut the y-axis at point (0,7), while the line \( y = 3x + 2 \) will slope upwards and cut the y-axis at point (0,2).

Identify the point of intersection

The intersection of these two lines will represent the solution of the system. By observing the graph, spot the point where these two lines intersect.

Interpret the graph

If, as in our case, the two lines intersect at one point then there is one unique solution to the system of equations. If the lines were parallel and never intersected, it would mean there is no solution. If the lines coincided fully, it would represent infinitely many solutions.