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Unit 4 Seminar Agenda Slope What it is, What it looks like, how to find it Ordered Pairs Types of...
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Transcript of Unit 4 Seminar Agenda Slope What it is, What it looks like, how to find it Ordered Pairs Types of...
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