Graphs of Linear Functions

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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
SLOPE-INTERCEPT FORM
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)
2)
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Graph the equation by hand.
3)
4)
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Graph the equation by hand.
5)
6)
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The only type of linear equation that CANNOT be
written in slope-intercept form is that of a
___________________ ___________. It is written
in the form of __________________.
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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 5.2x 2.2y 78.
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Finding the intercepts of a line graphically
(using the graphing calculator).
1) Solve the equation for y (do not round off
values) 2) Enter the equation into y1 3) To find
the x-intercept use ??? 2 Zero. 4) To find the
y-intercept hit ???.
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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

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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. 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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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.
TWO TYPES OF EQUATIONS WITHIN A SYSTEM 1)
Independent Equations Equations of a system that
have DIFFERENT graphs (For a system of 2
linear equations, this would mean two different
lines) 2) Dependent Equations Equations of a
system that have the SAME graph (For a system of
2 linear equations, this would mean the same
line)
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• From these options, three types of systems
  involving two equations in two variables can be
  formed.

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• 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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• 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

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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 One leaning on the
  other
• When the graphs of the two linear equations
  are the same, we say the lines coincide.
• Example

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Solve the system graphically (by hand).
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Solve the system graphically (using graphing
calculator).
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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
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Solve using the substitution method
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Solve using the substitution method
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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
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Solve using the elimination (addition) method
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Solve using the elimination (addition) method
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Solve using the elimination (addition) method
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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 Cost of Materials Cost of Labor
Racing 110 120
Mountain 140 180

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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
• Choose and define three variables to represent
  the unknown quantities.
• Using the info from the problem, write three
  equations involving the variables.
• Solve the system of equations using your
  calculator to find the reduced row-echelon form
  of the augmented matrix.
• State the solution to the problem in words.
  Include the appropriate units. Do NOT include the
  variables.

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Use a system of equations to solve the problem.
McDonalds 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?