Chapter 3 Chapter 3 Linear Equations and Linear Equations and

FunctionsFunctionsTSWBAT find solutions of two variable TSWBAT find solutions of two variable open sentences, and graph linear open sentences, and graph linear equations and points in two variables.equations and points in two variables.

Two Variable EquationsTwo Variable Equations

Open SentenceOpen Sentence – a statement or – a statement or equation that can have more than one equation that can have more than one solution. solution.

Open Sentence in Two VariablesOpen Sentence in Two Variables – A – A statement that can have more than one statement that can have more than one solution but contains two different solution but contains two different variables.variables.

Two Variable EquationsTwo Variable Equations

Solution Set Solution Set – Set of all solutions to a given – Set of all solutions to a given problem.problem.

Solution – Solution – To a two variable sentence is an To a two variable sentence is an Ordered Pair.Ordered Pair.

Ordered Pair – Ordered Pair – Solution to a two variable Solution to a two variable sentence or the value of the x and y terms to a sentence or the value of the x and y terms to a point on a graph. An ordered pair is written point on a graph. An ordered pair is written with the x-term first and the y-term second. with the x-term first and the y-term second. EX (X, Y); (2,5); (-5,3); (8,-1)EX (X, Y); (2,5); (-5,3); (8,-1)

GraphingGraphing

xy- Coordinate Planexy- Coordinate Plane – The normal two – The normal two directional plane on which an open directional plane on which an open sentence in two variables can be sentence in two variables can be graphed. graphed.

GraphingGraphing

Plane Rectangular Coordinate System Plane Rectangular Coordinate System or Cartesian coordinate system–or Cartesian coordinate system– Made Made up of two number lines that intersect at up of two number lines that intersect at right angles at the point right angles at the point OO. This is also . This is also named the Cartesian system after French named the Cartesian system after French mathematician Rene Descartes who mathematician Rene Descartes who introduced the idea of coordinates.introduced the idea of coordinates.

GraphingGraphing

OriginOrigin – The intersection point of the two – The intersection point of the two number lines in the coordinate system number lines in the coordinate system labeled as point labeled as point OO..

x-axisx-axis – The horizontal number line in a – The horizontal number line in a Coordinate System.Coordinate System.

y-axisy-axis – The vertical number line in a – The vertical number line in a Coordinate System.Coordinate System.

GraphingGraphing

Quadrants Quadrants – The four sections made up by the – The four sections made up by the intersection of the x and y axis in the coordinate intersection of the x and y axis in the coordinate system.system.

Plotting Plotting – In graphing points and lines on the – In graphing points and lines on the coordinate system we call graphing a point plotting the coordinate system we call graphing a point plotting the point or placing the point on the graph. point or placing the point on the graph.

DomainDomain – The values for which x can be in a two – The values for which x can be in a two variable sentence or on the coordinate system.variable sentence or on the coordinate system.

RangeRange – The values for which y can be in a two – The values for which y can be in a two variable sentence or on the coordinate system.variable sentence or on the coordinate system.

GraphingGraphing

One – to – One Correspondence – One – to – One Correspondence – between ordered pairs and points on the between ordered pairs and points on the plane can be summarized as:plane can be summarized as:

1. There is exactly one point on the plane 1. There is exactly one point on the plane associated with each ordered pair.associated with each ordered pair.

2. There is exactly one ordered pair 2. There is exactly one ordered pair associated with each point on the plane.associated with each point on the plane.

GraphingGraphing

GraphGraph - of an open sentence in two variables - of an open sentence in two variables is the set of all points in the coordinate plane is the set of all points in the coordinate plane that satisfies the sentence.that satisfies the sentence.

Linear Theorem - Linear Theorem - The graph of every The graph of every equation of the form : equation of the form : Ax + By = CAx + By = C, when , when AA and and BB are not both 0, is a line. Similarly every are not both 0, is a line. Similarly every line in the coordinate plane is the graph of an line in the coordinate plane is the graph of an equation in this form. equation in this form.

Linear vs. Non-LinearLinear vs. Non-Linear

Linear – Forms a lineLinear – Forms a line Examples of linear: 5x + 3y = -8,Examples of linear: 5x + 3y = -8,

Non-Linear – Does not form a line.Non-Linear – Does not form a line. Examples of Non-linear: 2x +3yExamples of Non-linear: 2x +3y22 = 4, = 4,

xy = 2, xy = 2,

5

2

5 y

x

53

2 y

x

X and Y InterceptsX and Y Intercepts

To solve for x-InterceptTo solve for x-Intercept

1. Solve equation for X.1. Solve equation for X.

2. Substitute 0 in for y.2. Substitute 0 in for y.

3. Solve3. Solve To solve for y-InterceptTo solve for y-Intercept

1. Solve equation for y.1. Solve equation for y.

2. Substitute 0 in for x.2. Substitute 0 in for x.

3. Solve3. Solve

Graph a LineGraph a Line

It is best to have 3 points on the line, but It is best to have 3 points on the line, but you only need 2.you only need 2.

The easiest way is to graph the two The easiest way is to graph the two intercepts and then plot the third point intercepts and then plot the third point you are given or find to determine the you are given or find to determine the direction of the line.direction of the line.

ExamplesExamples

Finding Solutions to two variable Finding Solutions to two variable equationsequations

GraphingGraphing

Graphing Points and LinesGraphing Points and Lines

GraphingGraphing

Finding X and Y InterceptsFinding X and Y Intercepts

Chapter 3Chapter 3

TSWBAT Find Slope of a line, and TSWBAT Find Slope of a line, and graph a line given the slope and graph a line given the slope and point on the line.point on the line.

SlopeSlope

Slope of a Line L = Slope of a Line L = where where

Horizontal Line – Slope = 0Horizontal Line – Slope = 0 Vertical Line – No SlopeVertical Line – No Slope Coefficient – number or numerical factor Coefficient – number or numerical factor

in front of a variable. In the y-equals in front of a variable. In the y-equals equation the coefficient in front of the x-equation the coefficient in front of the x-term is the slope of the line. term is the slope of the line.

12

12

xx

yy

run

rise

)( 12 xx

Slope TheoremsSlope Theorems

Theorem 2 – The slope of the line Theorem 2 – The slope of the line

Ax + By = C where is .Ax + By = C where is .

Theorem 3 – Let Theorem 3 – Let PP(x(x11,y,y11) be a point and ) be a point and

mm a real number. There is one and only a real number. There is one and only one line one line LL through through PP having the slope having the slope mm. . An equation of An equation of LL is is y – yy – y1 1 = m(x – x= m(x – x11))..

)0( BB

A

Slope GeneralizationsSlope Generalizations

The slope of a line rises if The slope of a line rises if mm is positive. is positive. The slope of a line falls if The slope of a line falls if mm is negative. is negative. The larger is, the steeper the line is. The larger is, the steeper the line is. m

SlopeSlope

Examples – Finding SlopeExamples – Finding Slope

Chapter 3Chapter 3

TSWBAT find an equation of a line given TSWBAT find an equation of a line given the slope and a point on the line, given the slope and a point on the line, given two points, or given the slope and y-two points, or given the slope and y-intercept.intercept.

Equations for a LineEquations for a Line

Standard Form of the equation of a line is Standard Form of the equation of a line is Ax + By = CAx + By = C with A, B, and C being integers. with A, B, and C being integers.

There are two other forms for the equation of a There are two other forms for the equation of a line however.line however.

Point-Slope form – the equation is then Point-Slope form – the equation is then y – y1 = m(x – x1)y – y1 = m(x – x1). .

Slope-Intercept form – the equation is Slope-Intercept form – the equation is y = mx + by = mx + b..

Finding Equation of a Finding Equation of a LineLine

Examples – Standard FormExamples – Standard Form

Finding Equation of a Finding Equation of a LineLine

Examples – Point Slope FormExamples – Point Slope Form

Finding Equation of a Finding Equation of a LineLine

Examples – Slope Intercept FormExamples – Slope Intercept Form

Chapter 3Chapter 3

TSWBAT find equations of parallel and TSWBAT find equations of parallel and perpendicular lines, find linear functions and graph perpendicular lines, find linear functions and graph them and determine if relations are functions.them and determine if relations are functions.

Parallel LinesParallel Lines

Parallel Lines have the same slope and Parallel Lines have the same slope and never intersect.never intersect.

ExampleExample

Perpendicular LinesPerpendicular Lines

Have the opposite reciprocal slope of the Have the opposite reciprocal slope of the other line.other line.

These lines meet at only one point in a These lines meet at only one point in a 90 degree angle.90 degree angle.

ExampleExample

Functions and RelationsFunctions and Relations Function – a correspondence between Function – a correspondence between

two sets, D and R, that assigns to each two sets, D and R, that assigns to each member of D exactly one member of R. member of D exactly one member of R. (One to One Correspondence).(One to One Correspondence).

Example:Example: Domain of the Function – is the Set D.Domain of the Function – is the Set D. Example:Example: Range of the Function – is the Set R.Range of the Function – is the Set R. Example:Example:

Functions and RelationsFunctions and Relations

Values of a function – the members of the Values of a function – the members of the range assigned to a member of the domain.range assigned to a member of the domain.

Example: the function f assigns 2 the value 4.Example: the function f assigns 2 the value 4.

Functional notation – f(x)=C Example: f(2)=4Functional notation – f(x)=C Example: f(2)=4 Linear Functions – A function f that can be Linear Functions – A function f that can be

defined by the equation f(x)=mx+b where x, m, defined by the equation f(x)=mx+b where x, m, and b are real numbers, and the graph of f is and b are real numbers, and the graph of f is the graph of the line y=mx+b with slope m and the graph of the line y=mx+b with slope m and y-intercept b.y-intercept b.

ExampleExample::

Functions and RelationsFunctions and Relations

Constant Function – A function where Constant Function – A function where m=0 and is thus f(x)=b for all x.m=0 and is thus f(x)=b for all x.

Example:Example: Is this a Horizontal or Vertical Line?Is this a Horizontal or Vertical Line? Rate of change m = slope of a line =Rate of change m = slope of a line =

xinchange

xfinchange )(

Functions and RelationsFunctions and Relations

Relation – Any set of ordered pairs. A Relation – Any set of ordered pairs. A function is a relation but not all function is a relation but not all relations are functions. A relation can relations are functions. A relation can contain two or more ordered pairs contain two or more ordered pairs with the same x and/or y values. A with the same x and/or y values. A function can contain two or more function can contain two or more ordered pairs with the same y value ordered pairs with the same y value only.only.

Example:Example:

Functions and RelationsFunctions and Relations

Vertical - Line Test – a test to Vertical - Line Test – a test to determine if a given relation is a determine if a given relation is a function. This test says a relation is a function. This test says a relation is a function if and only if a vertical line function if and only if a vertical line intersects the graph of the relation at intersects the graph of the relation at most one time.most one time.

Example:Example:

Chapter 3Chapter 3

TSWBAT Solve systems of Linear TSWBAT Solve systems of Linear Equations by 1. Linear Combinations, 2. Equations by 1. Linear Combinations, 2. Substitution, 3 Graphing.Substitution, 3 Graphing.

Systems of Linear Systems of Linear EquationsEquations

A system of linear equations or linear A system of linear equations or linear system – system –

a set of linear equations in a set of linear equations in the the same two variables.same two variables.

Example Example

Solutions to a Systems of Solutions to a Systems of Linear EquationsLinear Equations

1. simultaneous solution - an ordered 1. simultaneous solution - an ordered pair that satisfies both equations at pair that satisfies both equations at their point of intersection.their point of intersection.

ExampleExample

Solutions to a Systems of Solutions to a Systems of Linear EquationsLinear Equations

2. the null set for two lines that are 2. the null set for two lines that are parallel.parallel.

ExampleExample

Solutions to a Systems of Solutions to a Systems of Linear EquationsLinear Equations

3. a line if the set of linear systems is 3. a line if the set of linear systems is a group of coinciding lines.a group of coinciding lines.

ExampleExample

Systems of Linear Systems of Linear EquationsEquations

Equivalent systems – systems of Equivalent systems – systems of linear equations that have the same linear equations that have the same solution set.solution set.

ExampleExample

Linear Combination – the addition of Linear Combination – the addition of two equations.two equations.

ExampleExample

Systems of Linear Systems of Linear EquationsEquations

Consistent system – a system that has Consistent system – a system that has at least one solution.at least one solution.

ExampleExample

Inconsistent system – a system with Inconsistent system – a system with no solution and lines that are no solution and lines that are inconsistent.inconsistent.

ExampleExample

Systems of Linear Systems of Linear EquationsEquations

Dependent system – a system that has Dependent system – a system that has an infinite number of solutions and an infinite number of solutions and the lines are coinciding.the lines are coinciding.

ExampleExample

Transformations Transformations

1.Replacing an equation by an 1.Replacing an equation by an equivalent expression. – That is equivalent expression. – That is multiplying each side of an equation multiplying each side of an equation by the same non-zero number.by the same non-zero number.

TransformationsTransformations

2. Substituting for one variable in 2. Substituting for one variable in an equation for that variable obtained an equation for that variable obtained from another equation in the system.from another equation in the system.

TransformationsTransformations

3. Replacing any equation by the 3. Replacing any equation by the sum of that equation and another sum of that equation and another equation in the system. – That is add equation in the system. – That is add left sides, right sides, and then equate left sides, right sides, and then equate the results.the results.

Systems of Linear Systems of Linear EquationsEquations

Three methods to solveThree methods to solve 1. Linear Combination1. Linear Combination 2. Substitution2. Substitution 3. Graphing3. Graphing

Systems of Linear Systems of Linear EquationsEquations

Example Linear CombinationExample Linear Combination

Systems of Linear Systems of Linear EquationsEquations

Example SubstitutionExample Substitution

Systems of Linear Systems of Linear EquationsEquations

Example GraphingExample Graphing

Chapter 3Chapter 3

TSWBAT solve linear inequalities TSWBAT solve linear inequalities and systems of linear inequalities.and systems of linear inequalities.

Linear InequalitiesLinear Inequalities

Linear Inequality in Two Variables – is Linear Inequality in Two Variables – is when the equals sign in a linear equation when the equals sign in a linear equation in two variables is replaced by an in two variables is replaced by an inequality symbol like <, >, or inequality symbol like <, >, or

Boundary – The linear equation from Boundary – The linear equation from which the inequality was formed.which the inequality was formed.

Solution – a shaded region defined by Solution – a shaded region defined by the inequality symbol and boundary.the inequality symbol and boundary.

, .

Linear InequalitiesLinear Inequalities

Open Half-Plane – When the boundary Open Half-Plane – When the boundary line is not included in the solution and is line is not included in the solution and is shown as a dashed line (when we have < shown as a dashed line (when we have < or >).or >).

Closed Half-Plane – When the boundary Closed Half-Plane – When the boundary line is included in the solution and is line is included in the solution and is shown as a solid line (when we have shown as a solid line (when we have or ).or ).

Linear InequalitiesLinear Inequalities

Example - Example -

System of Linear System of Linear InequalitiesInequalities

System of Inequalities – Two or more System of Inequalities – Two or more linear inequalities working together as a linear inequalities working together as a set.set.

Solution to a system of Inequalities – is Solution to a system of Inequalities – is the region where ALL inequalities have a the region where ALL inequalities have a shaded region as a solution.shaded region as a solution.

System of Linear System of Linear InequalitiesInequalities

ExampleExample