Solve Special Types of Linear Systems

Inconsistent System a linear system with no solution. The graph of an inconsistent system resembles 2 parallel lines.

Consistent Dependent System a linear system with infinitely many solutions. The graph of the systems is the same line.

A linear system with no solution: Example: show that the linear system has no solution.

3x+2y=10 3x+2y=2

Put both equations in slope-intercept form Graph the linear systems- the linear are parallel because they have the same slope but different y-intercepts. Parallel lines do not intersect, so the system has no solution.

A linear system with infinitely many solutions Example: show that the linear system has infinitely many solutions.

x-2y=-4 1 y = x + 2 2 Put the firs linear system in slope-intercept form.

You will see that the linear equations are identical so they are the same line. So the linear system has infinitely many solutions.

Help in identifying the number of solutions.

Number of Solutions Soles and y-intercepts One Solution Different Slopes No Solution Same Slope Different y-intercepts Infinitely Many Solutions Same Slope Same y-intercept Identifying the Number of Solutions

Without solving the linear system, tell whether the linear system has one solution, no solution, or infinitely many solutions. a) 5x+y=-2 b) 6x+2y=3 -10x-2y=4 6x+2y=-5 Solution: Solution: Get both equations in slope-intercept Get both equations in slope-intercept form: form: Y=-5x-2 3 y = - 3x + Y=-5x-2 2 Because the lines have the same slope 5 y = - 3x - and the same y-intercept, the system has 2 infinitely many solutions. Because the lines have the same slope

but different y-intercepts, the system has no solution.