1. Linear Equations and Straight Lines
Chapter 1
Linear Equations and Straight Lines
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

2. Outline 1.1 Coordinate Systems and Graphs 1.2 Linear Inequalities
1.3 The Intersection Point of a Pair of Lines
1.4 The Slope of a Straight Line
1.4 The Slope of a Straight Line
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

3. Coordinate Plane
Origin O
x-axis
y-axis x y
Construct a Cartesian coordinate system on a plane by drawing two coordinate lines, called the coordinate axes, perpendicular at the origin. The horizontal line is called the x-axis, and the vertical line is the y-axis.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

4. Coordinate Plane: Points
Each point of the plane is identified by a pair of numbers (a,b). The first number tells the number of units from the point to the y-axis. The second tells the number of units from the point to the x-axis.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

5. Example Coordinate Plane
Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3). 3 -1
(-1,3) y
(2,1) 2 1 x -2 -1
(-1,-2)
(0,-3) -3
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

6. Example Graph of an Equation
Sketch the graph of the equation y = 2x - 1. x y
(-2,-5)
(-1,-3)
(0,-1)
(1,1)
(2,3) x y -2
2(-2) - 1 = -5 -1
2(-1) - 1 = -3
2(0) - 1 = -1 1
2(1) - 1 = 1 2
2(2) - 1 = 3
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

7. cx + dy = e (c, d, e constants)
Linear Equation
An equation that can be put in the form
cx + dy = e (c, d, e constants)
is called a linear equation in x and y.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

8. Standard Form of Linear Equation
The standard form of a linear equation is
y = mx + b (m, b constants)
if y can be solved for, or
x = a (a constant)
if y does not appear in the equation.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

9. Example Standard Form
Find the standard form of 8x - 4y = 4 and 2x = 6.
(a) 8x - 4y = 4
(b) 2x = 6
8x - 4y = 4
- 4y = - 8x + 4
y = 2x - 1
2x = 6
x = 3
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

10. Intercepts
x-intercept: a point on the graph that has a y-coordinate of 0. This corresponds to a point where the graph intersects the x-axis.
y-intercept: the point on the graph that has a x-coordinate of 0. This corresponds to the point where the graph intersects the y-axis.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

11. Graph of y = mx + b To graph the equation y = mx + b:
1. Plot the y-intercept (0,b).
2. Plot some other point. [The most convenient choice is often the x-intercept.]
3. Draw a line through the two points.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

12. Example Graph of Linear Equation
Use the intercepts to graph y = 2x - 1.
x-intercept: Let y = 0
0 = 2x - 1
x = 1/2
y-intercept: Let x = 0
y = 2(0) - 1 = -1 x y
(1/2,0)
(0,-1)
y = 2x - 1
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

13. 1.2 Linear Inequalities Definitions of Inequality Signs
Inequality Property 1
Inequality Property 2
Standard Form of Inequality
Graph of x > a or x < a
Graph of y > mx + b or y < mx + b
Graphing System of Linear Inequalities
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

14. Graph of x > a or x < a
The graph of the inequality
x > a consists of all points to the right of and on the vertical line x = a;
x < a consists of all points to the left of and on the vertical line x = a.
We will display the graph by crossing out the portion of the plane not a part of the solution.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

15. Example Graph of x > a
Graph the solution to 4x > -12.
First write the equation in standard form.
4x > -12
x > -3 x y
x = -3
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

16. Graph of y > mx + b or y < mx + b
To graph the inequality, y > mx + b or
y < mx + b:
1. Draw the graph of y = mx + b.
2. Throw away, that is, “cross out,” the portion of the plane not satisfying the inequality. The graph of y > mx + b consists of all points above or on the line. The graph of y < mx + b consists of all points below or on the line.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

17. Example Graph of y > mx + b
Graph the inequality 4x - 2y > 12.
First write the equation in standard form.
4x - 2y > 12
- 2y > - 4x + 12
y < 2x - 6
y = 2x - 6 y x
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

18. Example Graph of System of Inequalities
Graph the system of inequalities
The system in standard form is
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

19. 1.3 The Intersection Point of a Pair of Lines
Solve y = mx + b and y = nx + c
Solve y = mx + b and x = a
Supply Curve
Demand Curve
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

20. Solve y = mx + b and y = nx + c
To determine the coordinates of the point of intersection of two lines
y = mx + b and y = nx + c
Set y = mx + b = nx + c and solve for x. This is the x-coordinate of the point.
Substitute the value obtained for x into either equation and solve for y. This is the y-coordinate of the point.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

21. Example Solve y = mx + b & y = nx + c
Solve the system
Write the system in standard form, set equal and solve.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

22. Example Point of Intersection Graphy
y = 2x - 9/2
y = (-2/3)x + 7/3
(41/16,5/8)
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

23. 1.4 The Slope of a Straight Line
Slope of y = mx + b
Geometric Definition of Slope
Steepness Property
Point-Slope Formula
Perpendicular Property
Parallel Property
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

24. Slope of y = mx + b For the line given by the equation y = mx + b,
the number m is called the slope of the line.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

25. Example Slope of y = mx + b
Find the slope.
y = 6x - 9
y = -x + 4
y = 2
y = x
m = 6
m = -1
m = 0
m = 1
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

26. Geometric Definition of Slope
Geometric Definition of Slope Let L be a line passing through the points (x1,y1) and (x2,y2) where x1 ≠ x2. Then the slope of L is given by the formula
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

27. Example Geometric Definition of Slope
Use the geometric definition of slope to find the slope of y = 6x - 9.
Let x = 0. Then y = 6(0) - 9 = -9.
(x1,y1) = (0,-9)
Let x = 2. Then y = 6(2) - 9 = 3.
(x2,y2) = (2,3)
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

28. Steepness Property
Steepness Property Let the line L have slope m. If we start at any point on the line and move 1 unit to the right, then we must move m units vertically in order to return to the line. (Of course, if m is positive, then we move up; and if m is negative, we move down.)
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

29. Example Steepness Property
Use the steepness property to graph
y = -4x + 3.
The slope is m = -4.
A point on the line is (0,3).
If you move to the right 1
unit to x = 1, y must move
down 4 units to y = = -1.
y = -4x + 3
(0,3) x y
(1,-1)
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

30. Point-Slope Formula
Point-Slope Formula The equation of the straight line through the point (x1,y1) and having slope m is given by
y - y1 = m(x - x1).
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

31. Example Point-Slope Formula
Find the equation of the line through the point
(-1,4) with a slope of
Use the point-slope formula.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

32. Perpendicular Property
Perpendicular Property When two lines are perpendicular, their slopes are negative reciprocals of one another. That is, if two lines with slopes m and n are perpendicular to one another, then
m = -1/n.
Conversely, if two lines have slopes that are negative reciprocals of one another, they are perpendicular.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

33. Example Perpendicular Property
Find the equation of the line through the point (3,-5) that is perpendicular to the line whose equation is 2x + 4y = 7.
The slope of the given line is -1/2.
The slope of the desired line is -(-2/1) = 2.
Therefore, y -(-5) = 2(x - 3) or
y = 2x – 11.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

34. Parallel Property
Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

35. Example Parallel Property
Find the equation of the line through the point (3,-5) that is parallel to the line whose equation is 2x + 4y = 7.
The slope of the given line is -1/2.
The slope of the desired line is -1/2.
Therefore, y -(-5) = (-1/2)(x - 3) or
y = (-1/2)x - 7/2.
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.

36. Graph of Perpendicular & Parallel Lines
y = 2x - 11
2x + 4y = 7
y = (-1/2)x - 7/2
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel Copyright © 2010 Pearson Education, Inc.