04 linear equations in two variables

Linear Equations in Two Variables

Linear Equations in Two Variables

Cartesian Coordinate System

The rectangular or Cartesian coordinate system consists of a horizontal number line, the x-axis, and a vertical number line, the y-axis. The intersection of the axes is the origin.

Cartesian Coordinate System

The axes divide the coordinate plane, or the xy-plane, into four regions called quadrants.

The quadrants are numbered counterclockwise and they do not include any points on the axes.

Cartesian Coordinate System

Just as every real number corresponds to a point on the number line, every pair of real numbers corresponds to a point in the rectangular coordinate system.

Locating a point in the rectangular coordinate system that corresponds to a pair of real numbers is referred to as plotting or graphing the point.

Cartesian Coordinate System

Graph the points corresponding to the following pairs:(2, 4)(4, 2)(-2, -3)(-1, 3)(0, -4)(4, -2)

Cartesian Coordinate System

A pair of numbers, such as (2, 4), is called an ordered pair because the order of the numbers is important.

The pairs (4, 2) and (2, 4) correspond to different points.

The first number in an ordered pair is the x-coordinate and the second number is the y-coordinate.

X - Intercept

An x-intercept is a point on the x-axis. An x-intercept has a y-coordinate of 0.

Y - Intercept

A y-intercept is a point on the y-axis.A y-intercept has an x-coordinate of 0.

Exercises:

I. Plot the following points in a rectangular coordinate system. For each point, name the quadrant in which it lies or the axis on which it lies.(2, 5)3. (-5, 1)5. (2, -6)(-1, -6)4. (0, 4) 6. (3, 0)II.Explain why the point (0, -2) is not located in Quadrant IV.Explain why the point (-4, 0) is not located in Quadrant II.

Distance and Midpoint Formulas

The Distance Formula

The distance, d, between the points (x1, y1) and (x2,y2) in the rectangular coordinate system is

Example

Find the distance between (-1, 2) and (4, -3).

Solution Letting (x1, y1) = (-1, 2) and (x2, y2) = (4, -3), we obtain

11

The Midpoint Formula

Consider a line segment whose endpoints are (x1, y1) and (x2, y2).

To find the midpoint, take the average of the two x-coordinates and of the two y-coordinates. The coordinates of the segment's midpoint are

Example

Find the midpoint of the line segment with endpoints (1, -6) and (-8, -4).

Solution

(-7/2, -5) is midway between the points (1, -6) and (-8, -4).

Exercises:

Find the midpoint and length of the line segment with the given endpoints. (0, 0) and (6, 8)(-1, 0) and (6, -2)(-2, -5) and (5, 1)(1/2, 7) and (5, 2/5)(2, 4) and (0, 6)(-3, 5) and (3, -3)(0.2, 1.5) and (9.1, 6.4)

Linear Equations in Two Variables

Definition:

Let A, B, and C be real numbers such that A and B are not both zero. Then, an equation that can be written in the form:Ax + By = Cis called a linear equation in two variables.

Graphing a Linear Equations

Definition:

A solution to a linear equation is an ordered pair (x, y) that makes the equation a true statement.

For y = 3x 6, consider x = -1, 0,1, 2, 4. y = 3x 6 ordered pair solutions-9 = 3(-1) 6 (-1, -9)-6 = 3(0) 6 (0, -6) -3 = 3(1) 6 (1, -3) 0 = 3(2) 6 (2, -0) 1 = 3(4) 6 (4, 1)Therefore, the ordered pairs are solutions to the linear equation y = 3x 6.

Definition:

The Graph of an Equation in Two VariablesThe graph of an equation in two variables is the graph of all ordered pair solutions to the equation.

Graph: y=3x 6.

Graph: 3x + 2y = 12.

Solution:Rewrite the equation, 3x +2y = 12 2y = -3x + 12 y = -3/2 x + 6Arbitrarily select some values for x and find the corresponding y-values. Consider x = 0, 2, 4.-3/2 (0) + 6 = 6 (0, 6)-3/2 (2) + 6 = 3 (2, 3)-3/2 (4) + 6 = 0 (4, 0)

Graph: 3x + 2y = 12.

Solution:Plot the points and draw a line through them.(0, 6), (2, 3), (4, 0)

Graph: y = -2.

Solution:The equation y = -2 may be taken as y = 0x 2. No matter what the value for x is, y will always be equal to -2. The graph of such equation is a horizontal line passing through the point y = -2.

Graph: x =1.

Solution:The equation x = 1 may be taken as 0y + x = 1. No matter what the value for y is, x will always equal to 1. The graph of such equation is a vertical line passing through the point x = 1 in this case.

Using Intercepts for Graphing

If a line has distinct x- and y-intercepts, then they can be used as two points that determine the location of the line. Since horizontal lines, vertical lines, and lines through the origin do not have two distinct intercepts, they cannot be graphed using only the intercepts.

Example: Graph x + y = 5

Find the x intercept: (x, 0)y = 0: x + (0) = 5(5, 0)Find the y intercept: (0, y)x = 0:(0) + y = 5(0, 5)Graph the equation.

Try this!

Graph the following: y x = 1y = 2x

Exercises:

Graph the following:x + 2y = 42x - 3y = 6x + 5 = 0y + 1 = 06x + 3y = 0

Slope of a Line

Slope of a Line

The slope of a line is the ratio of the change in y-coordinate, or the rise, to the change in x-coordinate, or the run, between two points on the line.

Finding the slope from a graph

Find the slope of the line by going from point A to point B. A is located at (0, 3) and B at (2, 0). In going from A to B, change in y: -3 (going down) change in x: 2 (going right) So, the slope is m = -3/2

Finding the slope from a graph

Find the slope of the line by going from point A to point B. A is located at (2, 1) and B at (6, 3). In going from A to B, change in y: 2 (going up) change in x: 4 (going right) So, the slope is m = 2/4 =

Finding the slope from a graph

Find the slope of the line by going from point A to point B. A is located at (0,0) and B at (-6, -3). In going from A to B, change in y: -3 (going down) change in x: -6 (going left) So, the slope is m = -3/-6 =

Finding the slope from coordinates

A is located at (0, 3) and B at (2, 0).

Finding the slope from coordinates

A is located at (2, 1) and B at (6, 3).

Finding the slope from coordinates

A is located at (0,0) and B at (-6, -3).

Types of Slope

Positive Slope

Lines that increase, or rise, from left to right have a positive slope.

Negative Slope

Lines that decrease, or fall, from left to right have a negative slope.

Zero Slope

Horizontal lines have a slope of zero.

Undefined Slope

Vertical lines have an undefined slope.

Exercises:

Find the slope of the line that contains each of the following points:(2, 6) and (5, 1)(-3, -1) and (4, 3)(-2, -2) and (-1, 7)(2, 6) and (1, 8)(5, 1) and ( 5, -2)(0, 3) and ( -1, 3) (24.3, 11.9) and (3.57, 8.40)(-2.7, 19.3), (5.46, -3.28)

Equation of a Line

Equation of a Line

General Form

ax + by + c = 0

wherein a, b, and c are real numbers

Slope-Intercept Form

y = mx + b

wherein m is the slope of the line, and b is the y-intercept

Point-Slope Form

y y1= m (x x1)

wherein m is the slope of the line, and P(x1, y1) is a point on the line

Example:

Find an equation for the line through (-2, 5) and slope -3.Solution:

Example:

Find the equation of the line through the given pair of points (3,5) and (4,7).Solution:Find the slopeUse the slope and one point, say (3, 5) in the point-slope form

EXERCISES

Find the slope of the line that passes through (2,3) and (4,3).Find the slope and the y-intercept of 3x + 5y - 9 = 0.What is the slope of the line y - 4x + 6 = 0?What is the y-intercept of the line 3x + 2y = 5?What is an equation of the line through (4,1) and (2,4) ?What is an equation of the line passing through the points ( 6, -3 ) and ( -2, 3 )?

EXERCISES

Find an equation of the line which has a slope of 2/3 and a y-intercept of 2?Find an equation of the line with x-intercept -2 and y-intercept 2? What is an equation of the line through (-8,1) with undefined slope ?What is an equation of the line through (4,3) with slope equal to zero?

EXERCISES

For each of the given equations, do the following:Rewrite the equation in slope-intercept formDetermine the slope.Find the intercepts (x and y).Graph the equation.y 5x 10 = 02y x + 4 = 0-2x + y + 8 = 03y + 2x + 6 = 05x 5y 15 = 0

Finding Slope in an Application

EXERCISES

Determine the slope of the ramp up the stairs.

Determine the slope of the aircrafts takeoff path.

Determine the slope of the roof.

Applications of Slope:

Rate of Change

Example:

Worldwide carbon dioxide (CO2) emissions have increased from 14 billion tons in 1970 to 26 billion tons in 2000 (World Resources Institute, www.wri.org).

Example:

Find and interpret the slope of the line. Solution: (1970, 14) and (2000, 26)

CO2 emissions are increasing 0.4 billion tons per year.

Example:

b) Use the slope to predict the amount of worldwide CO2 emissions in 2010. Solution: (1970, 14), (2000, 26), (2010, ?)

If the CO2 emissions keep increasing 0.4 billion tons per year, then in 10 years the level will go up 10(0.4) or 4 billion tons.

So in 2010 CO2 emissions will be 26 +4 = 30 billion tons.

Lets try this!

The list price for a new Ford Crown Victoria four-door sedan was $21,135 in 1998 and $24,513 in 2004 (www.edmunds.com).

a) Find the slope of the line shown in the accompanying figure.b) Use the accompanying figure to predict the price in 2011.c) Use the slope to predict the price in 2011.