Unit 4 Seminar Agenda

• Slope What it is, What it looks like, how to find it Ordered Pairs

• Types of Lines Diagonal, Horizontal, and Vertical Parallel and Perpendicular

• Finding the Equation of a Line• Solving Systems of Equations

Graphing Substitution Addition/Elimination Consistent/Inconsistent, Dependent/Independent

The SLOPE of a Line

• How STEEP a line is• How DIAGONAL a line is• How big of an angle a line makes with the x – axis• A comparison of the vertical change a line makes with the

horizontal change a line makes.

The SLOPE of a LineWill be POSITIVE if the diagonal line is heading uphill (increasing) as you

look at it from left to right

From top to bottom

y = 1.5x - 6

y = (13/2)x + 3/2

y = (1/10)x + 4

The SLOPE of a LineWill be NEGATIVE if the diagonal line is heading downhill (decreasing)

as you look at it from left to right

From top to bottom

y = -1.5x - 6

y = (-13/2)x + 3/2

y = (-1/10)x + 4

The SLOPE of a LineWill be ZERO if the line is a HORIZONTAL line ….

From top to bottom

y = 5

y = 5/2

y = -1.2

y = -7

NEVER cross the x – axis

They have NO x – intercept

Are of the form y = b

The SLOPE of a LineWill be UNDEFINED if the line is a VERTICAL line ….

NEVER cross the y – axis

They have NO y – intercept

Are of the form x = a

From left to right

x = -15/2

x = -3

x = 2.2

x = 6

The SLOPE of a Line

• To calculate the slope of a line algebraically

vertical change

horizontal change

riseslope m

run

1 1 2 2

2 1

2 1

vertical change

horizontal change

The slope of a line passing through the points (x , y ) and (x , y )

can be found by

riseslope m

run

y yym

x x x

This IS VERY EASY ARITHMETIC … LET IT BE EASY!

Find the slope of the line through the points

(2, 3) and (-4, 2)

• Substitute the given values into the formula

• Perform the arithmetic (reduce fractions if possible)

2 1

2 1

y yym

x x x

Remember … the slope formula is:

Find the slope of the line in the given graph

• Remember … the slope formula is:• Substitute the values into the

formula

• Perform the arithmetic (reduce fractions if possible)

2 1

2 1

y yym

x x x

Parallel Lines

• NEVER intersect.

• If two lines are parallel,

Their slopes are

EXACTLY EQUAL

• Two lines with

equal slopes are parallel.

Perpendicular Lines

• Intersect once. • If two lines are perpendicular, Their slopes are NEGATIVE RECIPROCALS• Two lines with negative reciprocal slopes are perpendicular.

Equations of a Lines

• The most useful equation of a line is the slope – intercept equation.

• y = mx + bm is the symbol for slope

b is the symbol for the y – intercept

• Given ANY linear equation, if we rearrange it so there are NO grouping symbols, all like terms are combined, AND one side as the y all alone … then we are in slope intercept form.

SLOPE – INTERCEPT Equation

• y = mx + b … once you get your equation in this form, then you KNOW the slope and you KNOW the y – intercept (no calculations required to find them!)

For example: y = 6x – 5

SLOPE = 6

Y – INTERCEPT (0, -5)

SLOPE – INTERCEPT Equation

• This also means if you know the slope and you know the y – intercept, then you can come up with the equation very easily.

• Given slope = -1/4 and (0, 1/3), find the equation in slope – intercept form.

y = mx + b …. so ….

To find the slope – intercept form of data (if the slope and y intercept are not just given to you)

• Find the SLOPEUse the formula (if given ordered pairs or the

graph)Use the definition of parallel or perpendicular

line

To find the slope – intercept form of data (if the slope and y intercept are not just given to you)

• Find the SLOPE Use the formula (if given ordered pairs or the graph) Use the definition of parallel or perpendicular line

• Find the EQUATIONChoose one of the given ordered pairsSubstitute the pair you choose and the slope

you found into the formula If you only have one ordered pair, then use itIf you have two pairs, choose one of them (it does

not matter which you choose!)

2 1 2 1( )y y m x x

Find the slope – intercept form of the equation passing through (-2, -2) and

perpendicular to -5x + y = 4

• Find the SLOPE: Not given directly to use … BUT … we are told, our line is perpendicular to -5x + y = 4. Let’s find the slope of this line, then use the definition of perpendicular lines to find our slope.-5x + y = 4y = 5x + 4 … slope of this line is 5 … our line is

perpendicular, so our line has a slope of -1/5

Find the slope – intercept form of the equation passing through (-2, -2) and

perpendicular to -5x + y = 4

• Find the SLOPE: -1/5• Find the EQUATION:

Choose one of the given ordered pairs Substitute the pair you choose and the slope you found into the formula

2 1 2 1( )y y m x x

Systems of Equations

• Definition: A system of equations Two or more equations With two or more unknownsThat may or may not contain a common

solution.

Systems of Equations

• Solutions to a system of equations A system of equations with TWO equations

can have ONE solution, NO solutions, or INFINITE solutions.

The solution to a system of equations is an ordered pair that satisfies ALL the equations of the system.

Methods Used to Solve Systems of Equations

• There are many different methods available. Regardless of the method you use, you will get the same answer.

Guessing Graphing

Substitution Addition/Elimination

Solving Systems of EquationsThe GUESSING Method

• Ineffective and inefficient …. Will not be discussed any more

Solving Systems of EquationsThe GRAPHING Method

• Graph the equations on the same graph (using the techniques from the previous unit)

Solving Systems of EquationsThe GRAPHING Method

Now think about this: if you graph two straight lines on the same grid, three possible pictures emerge:

1. The two lines INTERSECT

They have ONE point in common

There is ONE unique solution to the system, in the form (x, y)

(2.95, 3.84)

Solving Systems of EquationsThe GRAPHING Method

Now think about this: if you graph two straight lines on the same grid, three possible pictures emerge:

2. The two lines are PARALLEL to each other

They have NO points in common

There is NO SOLUTION to the system

Solving Systems of EquationsThe GRAPHING Method

Now think about this: if you graph two straight lines on the same grid, three possible pictures emerge:

3. The two lines are the SAME—called COINCIDENTAL lines

They have ALL points in common

There are INFINITELY MANY SOLUTIONS to the system

Solving Systems of EquationsThe SUBSTITUTION Method

• Rearrange one of the equations (it does not matter which one) to get the X or the Y all alone (it does not matter which one).

• Take the result and substitute it into the OTHER equation.

• Solve for the unknown in this new equation.• Substitute your result into one of the ORIGINAL

equations and solve for the unknown.• Write your solution as an ordered pair.

Solving Systems of EquationsThe SUBSTITUTION Method

• Rearrange one of the equations (it does not matter which one) to get the X or the Y all alone (it does not matter which one).

x + y = 6

3x + 4y = 9

Solving Systems of EquationsThe SUBSTITUTION Method

• Take the result and substitute it into the OTHER equation.

x + y = 6 6

3x + 4y = 9

3(6 - y) + 4y = 9

x y

Solving Systems of EquationsThe SUBSTITUTION Method

Solve for the unknown in this new equation.

3(6 - y) + 4y = 9

Solving Systems of EquationsThe SUBSTITUTION Method

• Substitute your result into one of the ORIGINAL equations and solve for the unknown.

y = -9 3x + 4(-9) = 9

Solving Systems of EquationsThe SUBSTITUTION Method

• Write your solution as an ordered pair.

x + y = 6

3x + 4y = 9

The solution for this

System of Equations

( , )

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• While the substitution method allowed

us to make an equation have only one variable by replacement, the elimination method allows us to do the same thing by actually getting rid of one variable (temporarily, of course).

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• The goal in this method is to get the

numbers in front of both x’s OR both y’s to be additive inverses of one another (same number, opposite signs).

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• The goal in this method is to get the numbers in front of both

x’s OR both y’s to be additive inverses of one another (same number, opposite signs).

• That way … when we ADD the equations together, one of the variables will be ELIMINATED.

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH

equations are in Ax + By = C form

2x - 3y = -13

5x - 12y = -46

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH equations are in Ax + By = C

form

• Find (because they already exist) or create (by multiplying) additive inverses of one variable. IT DOES NOT MATTER WHICH VARIBLE YOU ELIMINATE

2x - 3y = -13

5x - 12y = -46

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH equations are in Ax + By

= C form

• Find (because they already exist) or create (by multiplying) additive inverses of one variable.

• Add the equations together to eliminate one of the variables.

2x - 3y = -13

5x - 12y = -46

5(2x - 3y = -13) -10x + 15y = 65

2(5x - 12y = -46) 10x - 24y = -92

-10x + 15y = 65

10x - 24y = -92

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH equations are in Ax + By = C

form

• Find (because they already exist) or create (by multiplying) additive inverses of one variable.

• Add the equations together to eliminate one of the variables.

• Solve for the unknown in this new equation.

-10x + 15y = 65

10x - 24y = -92

-9y = -27

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH equations are in

Ax + By = C form• Find (because they already exist) or

create (by multiplying) additive inverses of one variable.

• Add the equations together to eliminate one of the variables.

• Solve for the unknown in this new equation.

• Substitute your result into one of the ORIGINAL equations and solve for the unknown.

y = 3

2x - 3(3) = -13 OR 5x - 12(3) = -46

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH equations are in Ax + By = C

form

• Find (because they already exist) or create (by multiplying) additive inverses of one variable.

• Add the equations together to eliminate one of the variables.

• Solve for the unknown in this new equation.

• Substitute your result into one of the ORIGINAL equations and solve for the unknown.

• Write your solution as an ordered pair

2x - 3y = -13

5x - 12y = -46

( , )

Solving Systems of EquationsThe ADDITION/ELIMINATION

Method• Make sure BOTH equations are in Ax + By = C form

• Find (because they already exist) or create (by multiplying) additive inverses of one variable. (It does not matter which variable you eliminate!)

• Add the equations together to eliminate one of the variables.

• Solve for the unknown in this new equation.

• Substitute your result into one of the ORIGINAL equations and solve for the unknown.

• Write your solution as an ordered pair

Vocabulary of Systems of Equations

Some other terminology comes into play when you’re dealing with systems of equations.

Vocabulary of Systems of Equations

Some other terminology comes into play when you’re dealing with systems of equations.

DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).

Vocabulary of Systems of Equations

Some other terminology comes into play when you’re dealing with systems of equations.

DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).

INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).

Vocabulary of Systems of Equations

Some other terminology comes into play when you’re dealing with systems of equations.

DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).

INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).

CONSISTENT: there is at least one solution to the system (intersecting lines have one and coincidental lines have infinitely many).

Vocabulary of Systems of Equations

Some other terminology comes into play when you’re dealing with systems of equations.

DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).

INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).

CONSISTENT: there is at least one solution to the system (intersecting lines have one and coincidental lines have infinitely many).

INCONSISTENT: there is no solution to the system (parallel lines).

Vocabulary of Systems of Equations

Some other terminology comes into play when you’re dealing with systems of equations.

DEPENDENT: the graph of two lines looks like you only graphed one (coincidental lines).

INDEPENDENT: the graph of two lines shows up as two lines (parallel and intersecting lines both qualify).

CONSISTENT: there is at least one solution to the system (intersecting lines have one and coincidental lines have infinitely many).

INCONSISTENT: there is no solution to the system (parallel lines).

Systems of Equations

• Solutions to a system of equations ONE SOLUTION: Independent, ConsistentNO SOLUTIONS: Independent, InconsistentINFINITE SOLUTIONS: Dependent,

Consistent