february 17, 2015

Parallel &Perpendicular Lines

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.

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?

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

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

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

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 ofthe line perpendicularto the one given

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.

Graph the inequality: y < 5x + 1

Graphing Systems of Inequalities(3)

Let’s start by graphing an inequality:

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.

21

3

45

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

3m 1b

4

3m 5b

Solid Below

Dashed

Above

Class Notes: Systems of Inequalities

15

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

2m 5b

3

1m 2b

Dashed Above

Dashed

Above

.

Solving 3x3 Systems of Equations

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)

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 Variables1. 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).

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 = 6x = 2

2 + 3(- 4) – z = -11

– z = -1z = 1

Plug all three into one of original equations to check.

.

Class Work 3.4:

Show all work on separate sheet of paper.