Solve each equation. Leave answers as fractions.

1. 5x – 3 = 18

2. 4(x – 1) + 2 = 15

3. 8x + 9 = 14x – 3

4. 12x + 5 = 18x

5. 3(2x + 1) = 5(x + 7)

x = 21/5

x = 17/4

x = 2

x = 5/6

x = 32

Points, Lines

& Planes

Definition – A point represents one location in space. It has no size and NO DIMENSION.

AB

It represents the simplest form of geometry.

Notation: Points are named by capital letters.

Example: Points A and B

A series of points can create a line. A line extends in two directions without endpoints. A line has only one dimension.

Notation: Lines can be denoted by using two points that lie on the line or by using a lower case letter.

Given any two points, you can draw exactly one line. You can draw an infinite amount of lines through one point.

m

Line m

B

A

AB or BA

A two dimensional figure that extends in both dimensions forever and has no thickness.

Notation: A plane is either named by one capital letter (like a point) or by at least three points (but no more than four points) that lie in the plane.

Plane M

Plane ABC

or Plane ABCD

A C

BD

Where have you seen points, lines, and planes before in math class?

The Coordinate Plane

Graphing Lines in slope-intercept form. Ex. y = 2x + 2

Plotting Points. Ex. (-2, 4)

Space is the set of all points. Space has three dimensions.

Collinear Points are points that lie on the same line.

A

B

CD

E

B, C, and D are noncollinear points.

A, C, and D are collinear points.

A

B

CD

E

Yes!! In fact any two points are collinear. We can always draw exactly

one line between two given points.

Are A and B collinear points?

Coplanar Points are points that lie on the same plane.

DB

A C

A, B, C, and D are coplanar points.

G HJ

K

K, J, G, and H are noncoplanar points.

The symbol for “to intersect” is We can find the intersection between sets of numbers.Example: Set A = {-2, 0, 2, 5, 6, 8}

Set B = {-3, -2, -1, 0, 6}

Find A B = {-2, 0, 6}

We can also find the Union of two sets. Find A B.

A B = {-3, -2, -1, 0, 2, 5, 6, 8}

The intersection of two figures is the set of points that are in both figures.

If two lines intersect, then they intersect at a point.

B

r

q

line q line r at point B.

If two planes intersect, then they intersect at a line.

M

Np

plane M plane N at line p.

B

C

E

G

A

D

H

F

1) Where does AB meet AE?

Point A

2) Where does ABCD BCGF?

BC

3) DC DA at

4) EF EA at

Point D.

Point E.

B

C

E

G

A

D

H

F

Through any three points there is at least one plane. Through any three non-collinear points, there

is exactly one plane.

If two points are in a plane, then the line that contains those points is also in that plane.

All collinear points are also coplanar.

However, coplanar points are not necessarily collinear.

A

BC

D

Through a line and a point not on that line, there is exactly one plane.

If two lines intersect then exactly one plane contains both of them.