Transcript pps

Chapter 5
Graphs of Linear Functions
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Section 5.2
Graphs of Linear Functions
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The steepness of a line is measured by the slope of the line.
The slope of a line is the ratio of vertical distance to
horizontal distance between two points on a line.
m=
(x2, y2)
(x1, y1)
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Examples: Find the slope of the line through the given points.
a) (8, 7) and (2, -1)
b) (-2.8, 3.1) and (-1.8, 2.6)
c) (5, -2) and (-1, -2)
d) (7, -4) and (7, 10)
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On calculator, look at the following graphs and describe.
Y1 = x
Y2 = -x
Y3 = 2x
Y4 = 0.25x
The slope-intercept form of a line: ______________________
Where m is ___________ and (0, b) is the _______________
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Graph the equation by hand.
1) y  0.4 x  3
2) y  7.2  x
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Graph the equation by hand.
3)
y  5.0
4) 3x  6 y  0
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Graph the equation by hand.
5)
x
 5 y
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6) 2 x  3 y  6
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The only type of linear equation that CANNOT be written in
slope-intercept form is that of a ___________________
___________.
The equation of _____________ lines are always written in
the form: ____________.
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Definition: x - intercept
The x-intercept of a graph is the points at which the
graph intersects the x-axis.
The y-coordinate is always ____.
The x-intercept is written in the form (
,
).
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Definition: y - intercept
The y-intercept of a graph is the points at which the
graph intersects the y-axis.
The x-coordinate is always ____.
The y-intercept is written in the form (
,
).
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Finding the intercepts of a line algebraically.
Given the equation of the line,
1) To find the x-intercept, set y = 0 and solve for x.
2) To find the y-intercept, set x = 0 and solve for y.
Always express each intercept as an ordered pair.
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Example
Algebraically, find the coordinates of the x- and y- intercepts of 5x
– 2y = 8.
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Finding the intercepts of a line graphically.
1) Solve the equation for y (do not round off)
2) Enter the equation into y1
3) Use [CALC] 2: Zero to find the x-intercept.
4) Hit  to find the y-intercept.
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Example
Use the calculator to find the coordinates of the x- and y- intercepts
of 3.6x – 2.1 y = 22.68.
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Section 5.3
Solving systems of two linear equations in two
unknowns graphically
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A system of linear equations consists of two or more
equations that share the same variables.
Question: How would you describe the solution of a
system of two linear equations in two variables?
Answer: ______________________________________
______________________________________________.
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Determine whether (2, -5) is a solution of the system
3x  y  11

x  2 y  7
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Graphically, a solution of two linear equations in two variables
is a point that is on both lines.
a.k.a. Point of Intersection!
To solve a system of linear equations graphically,
1) Sketch the graph of each line in the same coordinate
plane. Label the equation of each line.
2) Find the coordinates of the point of intersection.
3) Check your solution by substituting it back into BOTH of
the original equations. Remember, it must satisfy both of
the equations.
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Three Types of Systems
(Two equations in two variables)
I. Two types of systems:
1) Consistent System: A system of equations that has a solution
(at least one)
2) Inconsistent System: A system of equations that has NO
solution.
II. Two types of equations within a system:
1) Independent Equations: Equations of a system that have
DIFFERENT graphs (For a 2 x 2 system, this would mean two
different lines)
2) Dependent Equations: Equations of a system that have the
SAME graph (For a system of two equations, this would mean
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the same line.)
1. A consistent system of independent equations
This system will have exactly one solution (an ordered pair).
Graphically, it will look like two lines that intersect at exactly one
point.
Example:
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
y

x 8

2

y   1 x  5

4
2
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2. An inconsistent system of independent equations
This system will have NO solution.
Graphically, it will look like two parallel lines (they never
intersect). Notice that these lines have the same slope but
a different y-intercept.
Example:
 y  3x  5

 y  3x  4
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3. A consistent system of dependent equations
This system will have an infinite number of solutions.
Graphically, it will look like a single line.
(Think: "Dependent = leaning on one another")
When the graphs of the two linear equations are the same,
we say the lines coincide.
Example:
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
y   x  5
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
2 x  3 y  15
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Solve the system graphically (by hand).
3x  y  3

2 x  y  7
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Solve the system graphically (using graphing calculator).
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1

 y   2 x  2

2 x  3 y  5

2
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Section 5.4
Solving systems of two linear equations in two
unknowns algebraically
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Solving a system of two linear equations
using the SUBSTITUTION METHOD
1.
If it is not done already, solve one of the equations for
one of its variables (isolate one of the variables).
2.
Substitute the resulting expression for that variable into
the other equation and solve it.
3.
Find the value of the remaining variable by substituting
the value found in step 2 into the equation found in step
1.
4.
State the solution (an ordered pair).
5.
Check the solution in both of the original equations.
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Solve using the substitution method:
x  2 y  6

3x  y  10
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Solve using the substitution method:
2 x  y  21

4 x  5 y  7
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Solve using the substitution method:
3
 x2 y
2

0.6 x  0.4 y  0.4
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Solving a system of two linear equations using the
ELIMINATION (ADDITION) method
1.
Write both equations in standard form Ax + By = C
2.
If necessary, multiply one or both equations by some
number(s) in order to make the coefficient of one of the
variables opposite.
3.
Add the equations to eliminate one of the variables.
4.
Solve the equation for the remaining variable.
5.
Substitute the value into either of the original equations to find
the value of the variable.
6.
Check the solution in the original equations.
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Solve using the elimination (addition) method:
x  y  5

x  y  7
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Solve using the elimination (addition) method:
3x  4 y  25

 y  2x  2
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Solve using the elimination (addition) method:
2a  3b  7

5a  2b  1
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Solve using the elimination (addition) method:
0.16 x  0.08 y  0.32

 y  2x  4
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Using a matrix to solve a system of linear
equations using the graphing calculator (RREF)
Refer to handout.
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Use a system of equations to solve the problem.
A bicycle manufacturer builds racing bikes and mountain bikes,
with per unit manufacturing costs as shown in table below. The
company has budgeted $31, 800 for labor and $26,150 for
materials. How many bicycles of each type can be built?
Model
Racing
Mountain
Cost of
Materials
$110
$140
Cost of Labor
$120
$180
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Use a system of equations to solve the problem.
$9500 is invested into two funds, one paying 8% interest and the
other paying 10% interest. The interest earned after one year was
$870. How much was invested in each fund?
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Systems of THREE linear equations
For the following problem:
1.
Choose and define three variables to represent the unknown
quantities.
2.
Using the info from the problem, write three equations
involving the variables.
3.
Solve the system of equations using your calculator to find the
reduced row-echelon form of the augmented matrix.
4.
State the solution to the problem in words. Include the
appropriate units. Do NOT include the variables.
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McDonald’s recently sold small soft drinks for $0.87, medium soft
drinks for $1.08, and large soft drinks for $1.54. During a lunchtime
rush, Chris sold 40 soft drinks for a total of $43.40. The number of
small and large drinks, combined, was 10 fewer than the number of
medium drinks. How many drinks of each size were sold?
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