Chapter 1 linear equations and straight lines

Chapter 1

Linear Equations and Straight Lines

Outline

1.1 Coordinate Systems and Graphs1.2 Linear Inequalities1.3 The Intersection Point of a Pair of Lines1.4 The Slope of a Straight Line1.5 The Method of Least Squares

Section 1.1

Coordinate Systems and Graphs

Outline

1. Coordinate Line

2. Coordinate Plane

3. Graph of an Equation

4. Linear Equation

5. Standard Form of Linear Equation

6. Graph of x = a

7. Intercepts

8. Graph of y = mx + b

Coordinate Line

Construct a Cartesian coordinate system on a line by choosing an arbitrary point, O (the origin), on the line and a unit of distance along the line. Then assign to each point on the line a number that reflects its directed distance from the origin.

Example Coordinate Line

Graph the points -3/5, 1/2 and 15/8 on a coordinate line.

-4 -3 -2 -1 0 1 2 3 4

Origin

-3/5 1/2 15/8

Unit length

Positive numbersNegative numbers

Origin

x-axis

y-axis

Coordinate Plane

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.

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.

Example Coordinate Plane

Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3).

(2,1)2

(0,-3)

-1(-1,3)

(-1,-2)

Graph of an Equation

The collection of points (x,y) that satisfies an equation is called the graph of that equation. Every point on the graph will satisfy the equation if the first coordinate is substituted for every occurrence of x and the second coordinate is substituted for every occurrence of y in the equation.

Example Graph of an Equation

Sketch the graph of the equation y = 2x - 1.

-2 2(-2) - 1 = -5

-1 2(-1) - 1 = -3

0 2(0) - 1 = -1

1 2(1) - 1 = 1

2 2(2) - 1 = 3

(-2,-5)

(-1,-3)

(0,-1)

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.

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.

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 = 6x = 3

Graph of x = a

The equation x = a graphs into a vertical line a units from the y-axis.

yx = 2

x = -3

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.

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.

Example Graph of Linear Equation

Use the intercepts to graph y = 2x - 1.x-intercept: Let y = 00 = 2x - 1x = 1/2

y-intercept: Let x = 0y = 2(0) - 1 = -1

(1/2,0)

(0,-1)

y = 2x - 1

Summary Section 1.1

Cartesian coordinate systems associate a number with each point of a line and associate a pair of numbers with each point of a plane.

The collection of points in the plane that satisfy the equation ax + by = c lies on a straight line. After this equation is put into one of the standard forms y = mx + b or x = a, the graph is easily drawn.

Section 1.2

Linear Inequalities

Outline

1. Definitions of Inequality Signs

2. Inequality Property 13. Inequality Property 24. Standard Form of

Inequality5. Graph of x > a or x < a6. Graph of y > mx + b or y <

mx + b 7. Graphing System of Linear

Inequalities

Definitions of Inequality Signs

a < b means a lies to the left of b on the number line.

a < b means a = b or a < b. a > b means a lies to the right of b on the

number line. a > b means a = b or a > b.

Which of the following statements are true?

-1 > -4

0 > -2

Inequality Signs Example

-4 -3 -2 -1 0 1 2 3 4

Inequality Property 1

Inequality Property 1 Suppose that a < b and that c is any number. Then a + c < b + c. In other words, the same number can be added or subtracted from both sides of the inequality.

Note: Inequality Property 1 also holds if < is replaced by >, < or >.

Example Inequality Property 1

Solve the inequality x + 5 < 2.Subtract 5 from both sides to isolate the x on the left.

x + 5 < 2x + 5 - 5 < 2 - 5

x < -3

The values of x for which the inequality holds are exactly those x less than or equal to −3.

Inequality Property 2

2A. If a < b and c is positive, then ac < bc.

2B. If a < b and c is negative, then ac > bc.

Note: Inequality Property 2 also holds if < is replaced by >, < or >.

Example Inequality Property 2

Solve the inequality -3x + 1 > 7.

Subtract 1 from both sides to isolate the x term on the left.

-3x + 1 > 7-3x + 1 - 1 > 7 - 1

-3x > 6Divide by -3, or multiply by -1/3 to isolate the x.

x < -2

Standard Form of Linear Inequality

A linear inequality of the form cx + dy < e

can be written in the standard form

1. y < mx + b or y > mx + b if d ≠ 0, or

2. x < a or x > a if d = 0.

Note: The inequality signs can be replaced by >, < or >.

Example Linear Inequality Standard Form

Find the standard form of 5x - 3y < 6 and 4x > -8.

(a) 5x - 3y < 6 (b) 4x > -8

5x - 3y < 6 -3y < - 5x + 6y > (5/3)x - 2

4x > -8x > -2

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.

Example Graph of x > a

Graph the solution to 4x > -12.

First write the equation in standard form.

4x > -12

x > -3

x = -3

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.

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

Example Graph of System of Inequalities

Graph the system of inequalities

The system in standard form is

2 3 15

4 2 12

Summary Section 1.2 - Part 1

The direction of the inequality sign in an inequality is unchanged when a number is added to or subtracted from both sides of the inequality, or when both sides of the inequality are multiplied by the same positive number. The direction of the inequality sign is reversed when both sides of the inequality are multiplied by the same negative number.

The collection of points in the plane that satisfy the linear inequality ax + by < c or

ax + by > c consists of all points on and to one side of the graph of the corresponding linear equation. After this inequality is put into standard form, the graph can be easily pictured by crossing out the half-plane consisting of the points that do not satisfy the inequality.

The feasible set of a system of linear inequalities (that is, the collection of points that satisfy all the inequalities) is best obtained by crossing out the points not satisfied by each inequality. The feasible set associated to the system of the previous example is a three-sided unbounded region.

Section 1.3

The Intersection Point of a Pair of Lines

Outline

1. Solve y = mx + b and y = nx + c

2. Solve y = mx + b and x = a

3. Supply Curve

4. Demand Curve

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

1. Set y = mx + b = nx + c and solve for x. This is the x-coordinate of the point.

2. Substitute the value obtained for x into either equation and solve for y. This is the y-coordinate of the point.

Example Solve y = mx + b & y = nx + c

Solve the system

Write the system in standard form, set equal and solve.

4 2 9.

2 7 92

1641 9 5

216 2 8

Example Point of Intersection Graph

Point of Intersection: (41/16, 5/8)

yy = 2x - 9/2

y = (-2/3)x + 7/3

(41/16,5/8)

Solve y = mx + b and x = a

To determine the coordinates of the point of intersection of two lines:

y = mx + b and x = a

1. The x-coordinate of the point is x = a.

2. Substitute x = a into y = mx + b and solve for y. This is the y-coordinate of the point.

Example Solve y = mx + b & x = a

Find the point of intersection of the lines

y = 2x - 1 and x = 2.

The x-coordinate of the point is x = 2.

Substitute x = 2 into y = 2x - 1

to get the y-coordinate.

y = 2(2) - 1 = 3

Intersection Point: (2,3)

y = 2x - 1

Supply Curve

For every quantity q of a commodity, the supply curve specifies the price p that must be charged for a manufacturer to be willing to produce q units of the commodity.

Supply Curve

Demand Curve

For every quantity q of a commodity, the demand curve gives the price p that must be charged in order for q units of the commodity to be sold.

Demand Curve

Example Supply = Demand

Suppose the supply and demand for a quantity is given by p = 0.0002q + 2 (p in dollars) and p = -0.0005q + 5.5. Determine both the quantity of the commodity that will be produced and the price at which it will sell when supply equals demand.

.0002 2 .0005 5.5

.0007 3.5

5000 units

.0002(5000) 2 $3

Summary Section 1.3

The point of intersection of a pair of lines can be obtained by first converting the equations to standard form and then either equating the two expressions for y or substituting the value of x from the form x = a into the other equation.

Section 1.4

The Slope of a Straight Line

Outline

1. Slope of y = mx + b

2. Geometric Definition of Slope

3. Steepness Property

4. Point-Slope Formula

5. Perpendicular Property

6. Parallel Property

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.

Example Slope of y = mx + b

Find the slope.

y = 6x - 9

y = -x + 4

m = -1

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

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) 3 9 12

62 0 2

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.)

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 = 3 - 4 = -1. y = -4x + 3

(1,-1)

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).

Example Point-Slope Formula

Find the equation of the line that passes through (-1,4) with a slope of .3

Use the point-slope formula.

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.

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.

Parallel Property

Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel.

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.

Graph of Perpendicular & Parallel Lines

2x + 4y = 7

y = (-1/2)x - 7/2

y = 2x - 11

The slope of the line y = mx + b is the number m. It is also the ratio of the difference between the y-coordinates and the difference between the x-coordinates of any pair of points on the line.

The steepness property states that if we start at any point on a line of slope m and move 1 unit to the right, then we must move m units vertically to return to the line.

The point-slope formula states that the line of slope m passing through the point (x1, y1) has the equation y - y1 = m(x - x1).

Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is –1.

Section 1.5

The Method of Least Squares

Outline

1. Least Squares Problem

2. Least Squares Error

3. Least Squares Line

4. Least Squares Using Technology

Least Squares Problem

Least Squares Problem Given observed data points (x1, y1), (x2, y2),…, (xN, yN) in the plane, find the straight line that “best” fits these points.

Least Squares Error

Least Squares Error Let Ei be the vertical distance between the point (xi, yi) and the straight line. The least-squares error of the observed points with respect to this line is

E = E12 + E2

2 +…+ EN2.

Example Least Squares Error

Determine the least-squares error when the line y = 1.5x + 3 is used to approximate the data points (1,6), (4,5) and (6,14).

Data Point Point on Line Vertical Distance Ei2

(1,6) (1, 4.5) 1.5 2.25

(4,5) (4,9) 4 16

(6,14) (6,12) 2 4

E = 22.25

Graph of Least Squares Error

(6,14)

Least Squares Line

Least Squares Line Given observed data points (x1, y1), (x2, y2),…, (xN, yN) in the plane, the straight line y = mx + b for which the error E is as small as possible is determined by

N xy x ym

y m xb

Example Least Squares Error

Find the least-squares line for the data points (1,6), (4,5) and (6,14).

x y xy x2

1 6 6 1

4 5 20 16

6 14 84 36

x = 11 y = 25 xy = 110 x2 = 53

Example Least Squares Error (2)

3 110 11 25 551.45

3 53 11 385525 1138 3.03

31.45 3.03

Least Squares Using Technology

Use Excel to find the least-squares line for the data points (1,6), (4,5) and (6,14).

y = 1.4474x + 3.0263

Summary Section 1.5

The method of least squares finds the straight line that gives the best fit to a collection of points in the sense that the sum of the squares of the vertical distances from the points to the line is as small as possible. The slope and y-intercept of the least-squares line are usually found with formulae involving sums of coordinates or by using technology.

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