Solve each equation. Leave answers as fractions. 1. 5x – 3 = (x – 1) + 2 = x + 9 = 14x – x + 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. A B 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 B D

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 C D E B, C, and D are noncollinear points. A, C, and D are collinear points.

A B C D 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. D B A C A, B, C, and D are coplanar points. GH J 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 N p 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.