2. Parallel &
Perpendicular Lines

3. Parallel Lines Have the Same Slope
• The lines never touch. Therefore:
• There is no solution to a system of equations
• If you know the slope of one line, you know the
slope of a line parallel to it.
• If you know the slope and a point on the line,
you can use the point-slope formula to find the
equation of, and graph the line.

4. The question says: The
equation of a line parallel to
the one shown could be;
It doesn’t matter what
points the line goes
through, as long as the
slope is the same
Find the slope..
A) y = 3x – 7 B) y < 3x – 7 C) 4x +2y = -11 D) 3x –y = 4
Which is the equation of a
line parallel to all those
shown on the graph?

5. Write the equation and graph the line parallel to
the one shown and passing through the point (3,-4).
Whole different
story, this one. The
line must pass thru
Again, find the slope of
the current line, then use
the point-slope formula
to find the new equation.
Point-Slope:
Y + 4 = -3(x – 3)
Equation:
Y = -3x + 5

6. A. y = -4/3x + 3 B. y = 4/3x -2 C. y
= -3/4x + 2 D. y = -4x -3
E. None
An equation of a line parallel to the graph could be:
Parallel Lines

7. What is the equation of the line parallel to the line
2y –x = 1, and passing thru the point (-4,5)
A. y = -x + 5/2 B. y = 2x – 5/2 C. y = -1/2x + 5
D. y = 1/2x – 5 E. None
Parallel Lines

8. Perpendicular Lines Have Slopes that are the
opposite inverse of each other
• The lines cross at a 90 degree angle
• There is always one solution
• If you know the slope of one line, change the sign
and use the reciprocal
• If you know the slope and a point on the line, you
can use the point-slope formula to find the
equation of, and graph the line.
Write the equation of
the line perpendicular
to the one given

9. Write the equation of the line perpendicular to
the one shown passing through the point (3,-4).
Again, find the slope of
the current line, then use
the point-slope formula
to find the new equation.

11. Graph the inequality: y < 5x + 1
Graphing Systems of Inequalities(3)
Let’s start by graphing an inequality:

12. Class Notes: Systems of Inequalities(3)
1. Write the equation in slope-intercept form.
2. Graph the y-intercept and slope.
3. Draw the line (solid or dashed).
 , Dashed line
 , Solid line
• Steps to Graphing Linear System Inequalities
,  Above y-intercept
,  Below y-intercept
4. Lightly shade above or below the y-intercept.
5. Graph the other equation. See #’s 3 and 4
6. Darkly shade overlap.

13. 2
1
3
4
5
3
y x
y x

 

   

Ex.
Graph the system of linear inequalities.
2) Graph.
Find m and b.
3) Solid or
dashed?
4) Lightly
shade above
or below the
y-intercept?
1) Put in
slope-intercept
form.
5) Do the same
for the other
equation.
6) Darkly
shade overlap.
2
3
m  1b  
4
3
m   5b 
Solid Below
Dashed
Above
Class Notes: Systems of Inequalities

14. 1
5
2
3 2
y x
y x

 

   
Graph the system of linear inequalities.
2) Find m and
b, then graph
3) Solid or
dashed?
4) Lightly
shade above
or below the
y-intercept?
1) Put in slope-
intercept form.
5) Do the same
for the other
equation.
6) Darkly
shade overlap.
1
2
m  5b 
3
1
m   2b  
Dashed Above
Dashed
Above

15. .
Solving 3x3 Systems of
Equations

16. Solving 3x3 Systems
A 3x3 system of equations has 3 unknown variables, and
therefore must have 3 equations.
We will look at two methods of solving 3x3 systems. The
method used depends entirely on the number of unknowns in
each equation.
A) Only 1 of the equations has all three variables in the
equation. This is the easier of the two. Let’s look:
Solve the System: 4x + 2y - z = -5
3y + z = -1
2z = 10
Begin at the bottom and
work your way up.
Plug z into 2nd equation
and solve for y.
Plug y and z into 1st
equation and solve for x.Plug all three in together
and check your solutions.
5. The solution set is (1, -2, 5)

17. Solving 3x3 Systems
B) All 3 of the equations contain all three variables in
the equation. Follow these steps to solve:
Steps for Solving in 3 Variables
1. Take the 1st 2 equations, cancel one of the variables.
2. Take the last 2 equations, cancel the same variable from step 1.
3. Take the results from steps 1 & 2 and use elimination solve
for both variables.
4. Plug the results from step 3 into one of the original 3
equations and solve for the 3rd remaining variable.
5. Write the solution as an ordered triple (x,y,z).

18. 1. Solve the system.
(2, -4, 1)
3 11
2 1
5 2 3 21
x y z
x y z
x y z
   
  
  



Solving 3x3 Systems
x + 3y – z = -11
+ 2x + y + z = 1
2x + y + z = 1
+ 5x – 2y + 3z = 21
Must eliminate
the z here also.
3x + 4y = -10 - x - 5y = 18
+ -x - 5y = 183( )
+ -3x -15y = 54
- 11y = 44
-3( )
y = - 4
3x + 4(-4) = -10
3x = 6
x = 2
2 + 3(- 4) – z = -11
– z = -1
z = 1
Plug all three into
one of original
equations to check.

19. .
Class Work 3.4:
Show all work on separate
sheet of paper.