1. Section 1-1 Points and Lines

2. Each point in the plane can be associated with an ordered pair of numbers, called the coordinates of the point. Each ordered pair of numbers can be associated with a point in the plane. To set up a coordinate system, we can choose two perpendicular lines. The x- axis is the horizontal line and the y-axis is the vertical line. Their point of intersection is called the origin. The axes divide the plane into four quadrants.

3. The diagram shows that P has x- coordinate 4 and y-coordinate -2. This is written as P(4, -2). Points with x- coordinate 0, such as R, lie on the y-axis. Points with y-coordinate 0, such as Q, lie on the x-axis. The origin, O, has coordinates (0, 0).

4. Linear Equations A solution of the equation 2x – 3y = 12 is an ordered pair of numbers that makes the equation true. (0, -4) is a solution because 2(0) – 3(-4) = 12. Several solutions are shown in the diagram on p. 1.

5. The set of all points in the plane corresponding to solutions of an equation is called the graph of the equation. We call -4 the y-intercept of the graph because the line intersects the y-axis at (0, -4). We call 6 the x-intercept of the graph because the line intersects the x-axis at (6,0).

6. Any equation of the form Ax + By = C, where A and B are not both 0, is called a linear equation because its graph is a line. Conversely, any line in the plane is the graph of a linear equation. We call Ax + By = C the general form of a linear equation.

7. ** When one of the constants A, B, or C in Ax + By = C is 0, you can draw certain conclusions about the graph. If C = 0, the line contains the origin. If A = 0, the line is horizontal. If B = 0, the line is vertical. Examine diagrams p. 2.

8. Intersection of Lines You can determine where two lines intersect by drawing their graphs or by solving their equations simultaneously. You can make hand-drawn sketches or you can use the graphing calculator to obtain the graphs.

9. Be aware that solutions found by graphing are not always exact; however an algebraic solution yields the exact values.

10. To solve equations simultaneously, you can multiply both sides of equation (1) by 3 and both sides of equation (2) by 2. Then subtract the second equation from the first equation.

11. Example Solve: 2x + 5y = 10 3x + 4y = 12

12. When two linear equations have no common solution their graphs are parallel lines. When two linear equations have infinitely many common solutions, the equations have the same graph. Examine diagrams p. 3

13. Example Solve 3x – y = 9 7x – 5y = 25 Then graph the equations.

14. Recall from Geometry… We denote the line segment with endpoints A and B as and its length as AB.

15. The Distance and Midpoint Formulas Let A =, B =, and M be the midpoint of. Then: AB = Distance Formula M = Midpoint Formula

16. Example Use A(2, 0), B(0, 8), C(-4, 7), and D(-2, -1). A. Show that and bisect each other. B. Show that AC = BC. C. What kind of figure is ABCD? D. Find the length of. E. Find the midpoint of.