5.3 System of Equation - Special Case

There are two ways to solve a system of equations with parallel lines:

1. Graphically: Graph the two equations on a coordinate plane. If the lines never intersect, then they are parallel.

2. Analytically: Solve for the x- and y-intercepts of each equation. If the x-intercepts are different, then the lines are parallel.

Graphical Solution 1. Set the equations equal to y to make graphing easier. 2. Identify the slope and y-intercept in each equation. 3. Graph the lines. If the two lines never intersect, then they are parallel. Analytical Solution

1. Solve for the x-intercept of each equation.

2. If the x-intercepts are different, then the lines are parallel.

For example, consider the following system of equations:

y = 2x + 1

y = 2x + 5

The slope of both equations is 2, and the y-intercepts are 1 and 5. Since the y-intercepts are different, the lines are parallel.

This can also be verified by graphing the equations. The two lines never intersect, so they are parallel.

Conclusion

A system of equations with parallel lines has no solution. This is because parallel lines never intersect. The two lines can be solved graphically or analytically.

here are two ways to solve a system of equations with perpendicular lines:

1. Graphically: Graph the two equations on a coordinate plane. If the lines intersect at a right angle, then they are perpendicular.

2. Analytically: Determine the slopes of the two equations. If the slopes are negative reciprocals of each other, then the lines are perpendicular.

Graphical Solution

1. Set the equations equal to y to make graphing easier.

2. Identify the slope and y-intercept in each equation.

3. Graph the lines.

If the two lines intersect at a right angle, then they are perpendicular.

Analytical Solution

1. Determine the slope of each equation.

2. If the slopes are negative reciprocals of each other, then the lines are perpendicular.

For example, consider the following system of equations:

y = 2x + 1

y = -1/2x + 3

The slope of the first equation is 2, and the slope of the second equation is -1/2. The slopes are negative reciprocals of each other, so the lines are perpendicular.

This can also be verified by graphing the equations. The two lines intersect at a right angle, so they are perpendicular.

Conclusion

A system of equations with perpendicular lines has exactly one solution. This is because perpendicular lines intersect at a single point. The two lines can be solved graphically or analytically.