Exercise 1 Page 182

If we are given a linear system with two unknowns, we can start by reviewing how systems of equations are classified in terms of the number of solutions they have.

From what we can see above, we can start by identifying if the system is inconsistent or not. Then, if it is consistent we can check if it is dependent or independent. Furthermore, remember that when solving a system of equations by graphing, the intersection point of the linear equations represents the solution of the system.

If we transform our system into slope-intercept form, we can classify the system by just looking at the y-intercepts and slopes of the equations forming it.

Relation between the Slopes and y-intercepts of the System	Number of Solutions of the System
Same slopes and same y-intercepts	The equations are equivalent, therefore, the system will have infinitely many solutions as the equations represent coinciding lines. This is a dependent system.
Same slopes and different y-intercepts	The equations represent parallel lines. The system has no solution. This is an inconsistent system.
Different slopes and same y-intercepts	The equations represent intersecting lines. The system has one solution. This is an independent system.
Different slopes and different y-intercepts	The equations represent intersecting lines. The system has one solution. This is an independent system.