Understanding how parallel and perpendicular lines work in systems of linear equations is important in 9th-grade Algebra I. This helps us see how two variables relate when we look at graphs.

Parallel Lines

Parallel lines go in the same direction, and they never meet. They have the same slope but different starting points, called y-intercepts.

For example, if we write two lines like this:

• Line 1: ( y = mx + b_1 )
• Line 2: ( y = mx + b_2 )

Here, ( m ) is the slope, and ( b_1 ) and ( b_2 ) are the y-intercepts. Since ( b_1 ) is not equal to ( b_2 ), that shows they are different lines.

Effect on Solutions

When we graph parallel lines, we can see that they never cross.

• Number of Solutions: This means there are no solutions where both equations are true at the same time because there isn't any point (x, y) that works for both.

For example, the lines ( y = 2x + 3 ) and ( y = 2x - 5 ) are parallel. There are many points that work for each line, but no point works for both lines at once.

Perpendicular Lines

Perpendicular lines meet at right angles. The special thing about them is that their slopes are negative reciprocals of each other.

If we have two lines like this:

• Line 1: ( y = m_1x + b_1 )
• Line 2: ( y = m_2x + b_2 )

Then we can say:

• Slopes: ( m_1 \cdot m_2 = -1 )

Effect on Solutions

When we graph perpendicular lines, we see that they cross at exactly one point.

• Number of Solutions: This point where they meet gives us exactly one unique solution that corresponds to the coordinates of the intersection.

For instance, if we have one line as ( y = 2x + 3 ) and another as ( y = -\frac{1}{2}x + 1 ), these lines are perpendicular. We can find their intersection by setting the equations equal:

1. Set the equations equal: ( 2x + 3 = -\frac{1}{2}x + 1 ).
2. Solve for ( x ): Combine terms to get ( 2.5x = -2 ).
3. Find ( x = -0.8 ), and plug it back in to find ( y = 2(-0.8) + 3 = 1.4 ).

So, the solution is the point ( (-0.8, 1.4) ).

Summary

• Parallel Lines: There are no solutions since the lines never cross.
• Perpendicular Lines: There is exactly one solution, found at their intersection point.

Understanding how to describe and analyze the relationships of lines helps us solve linear equations and creates a solid foundation for studying algebra and geometry in the future.