Writing and Graphing Linear Equations by yh1s7G

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									Linear Equations in Two
       Variables
   Graphing Linear Equations
 Writing Equations and Graphing

• These activities introduce rates of change and
 defines slope of a line as the ratio of the vertical
         change to the horizontal change.

 • This leads to graphing a linear equation and
 writing the equation of a line in three different
                      forms.
The Coordinate Plane
 Graphing ordered pairs in the
       coordinate plane
               Coordinate
                  axes


                       The coordinate plane is formed by
                          placing two number lines called
                         coordinate axes so that one is
                           horizontal (x-axis) and one is
origin   quadrants        vertical (y-axis). These axes
                          intersect at a point called the
                         origin (0, 0), which is labeled 0.
                        An ordered pair (x, y) is a pair of
                       real numbers that correspond to a
                       point in the coordinate plane. The
                        first number in an ordered pair is
                        the x-coordinate and the second
                           number is the y-coordinate.
           ordered
             pair
  In a previous lesson, you
learned to plot points from
   a table of values. The
 solution to an equation in
two variables is all ordered
pairs of real numbers (x, y)
 that satisfy the equation.
 You can graph the solution
    to an equation in two
variables on the coordinate
  plane by using the (x, y)
   values from the table.
 Graph each ordered pair
 in the table. Do all the
points appear to lie on the
            line?
Make a table of ordered pairs for each
 equation. Then graph the ordered
   pairs. (x = -3, -2, -1, 0, 1, 2, 3)

             1) y = x + 3
             2) y = x – 2
              3) y = 2x
              4) y = -2x
              5) y = - 2
             6) y = -x + 1
             7) y = x2 – 1
               8) y = 3
            9) y = 2x2 - 1
     In everyday life, one
  quantity often depends on
 another. On an automobile
trip, the amount of time that
you spend driving depends on
  your speed. In cases like
   these, it is said that the
second quantity is a function
  of the first. The second
quantity is called dependent,
       while the first is
         independent.

   Can you think of other
situations in real-life where
one quantity is dependent on
          another.
Many functions (equation)
can be represented by an
equation in two variables.
      The variable that
represents the dependent
    quantity is called the
 dependent variable. The
 variable that represents
the independent quantity
 is called the independent
variable. When you write
    an ordered pair using
these variables (x, y), the
    independent variable
        appears first.
(independent, dependent)
In the function y = 2x, each
     x-value is paired with
  exactly one y-value. Thus
  the value of y depends on
        the value of x.
In a function, the variable of
   the domain is called the
  independent variable and
 the variable of the range is
     called the dependent
  variable. On a graph, the
    independent variable is
      represented on the
    horizontal axis and the
     dependent variable is
 represented on the vertical
              axis.
          Example

 The value v (in cents) of n
   nickels is given by the
 function (equation) v = 5n

     a) Identify the
independent and dependent
         variable.

b) Represent this function
in a table for n = 0, 1, 2,
        3, 4, 5, 6.

c) Represent this function
        in a graph.
                   Use both a table and a graph to
Domain
                   represent each function for the
         Range     given values of the independent
                               variable.

                 1) The value v of 0, 1, 2, 3, 4, 5, and 6
                  dimes is a function of the number of
                                dimes, d.

                    2) The distance d traveled by a
                  jogger who jogs 0.1 miles per minute
                   over 0, 1, 2, 3, 4, and 5 minutes is a
                            function of time, t.
                                                   Try another

                 3) One hat cost $24. The total cost c
                 of 1, 2, 3, 4, 5, and 6 hats is a function
                   of the number n of hats purchased.
  A 1200-gallon tank is empty. A valve is
opened and water flows into the tank at the
  constant rate of 25 gallons per minute.

1) How would you find the amount of water in
  the tank after 5 minutes, 6 minutes, and 7
                   minutes?

 2) How would you find the time it takes for
 the tank to contain 500 gallons, 600 gallons,
               and 700 gallons?


           What is the independent variable?
            What is the dependent variable?
         What function represents this situation?
Slope and Rate of
     Change
Finding the slope of a Line
Slope – A measure of the steepness of a line on a
graph; rise divided by the run.
If P (x1, y1) and Q (x2, y2) lie along a nonvertical
line in the coordinate plane, the line has slope m,
given by




Linear equation – An equation whose solutions
form a straight line on a coordinate plane.
Collinear – Points that lie on the same line.
   Remember, linear equations have constant
slope. For a line on the coordinate plane, slope
   is the following ratio. This ratio is often
         referred to as “rise over run”.
       Slope and
Orientation of a Line
   In the coordinate
    plane, a line with
  positive slope rises
from left to right. A
   line with negative
 slope falls from left
to right. A line with 0
  slope is horizontal.
     The slope of a
     vertical line is
       undefined.
Find the slope of the line that passes through
             each pair of points.

             1)   (1, 3) and (2, 4)

             2)   (0, 0) and (6, -3)

             3)   (2, -5) and (1, -2)

             4)   (3, 1) and (0, 3)

             5)   (-2, -8) and (1, 4)
  Finding points on a Line
Graphing a line using a point
and the slope is shown here.

  Graph the line passing
through (1, 3) with slope 2.

    a) Graph point (1, 3).

 b) Because the slope is 2,
you can find a second point
on the line by counting up 2
units and then right 1 unit.

C) Draw a line through point
  (1, 3) and the new point.
  A line with slope –¼
   contains P(0, 5).

  a) Sketch the line

b) Find the coordinates
of a second point on the
          line.
Sketch each line from the information given.
      Find a second point on the line.

         a) point H(0, 0), slope 2
         b) point S(3, 3), slope -3
       c) point J( 2, 5); slope –1/2
       d) point T(-3, 1); slope – 2/5
  Finding Slope from a
         Graph

    Choose two points on
 the line (-4, 4) and (8, -
  2). Count the rise over
  run or you can use the
 slope formula. Notice if
you switch (x1, y1) and (x2,
   y2), you get the same
           slope:
    Real-world Applications
You have been solving problems
in which quantities change since
early school days. For instance,
  here is a typical first-grade
            problem:

  If I have two apples, and
then I buy five more apples,
how many apples do I have in
             all?

This situation concerns a change
    in the number of apples.


                   Try this problem
                    One marker on a road showed 20
                     miles. The next marker showed
                    170 miles. How many miles have
                                gone by?
                    The time was 2:00 pm at the first
                     marker. Then it was 5:00 pm at
                      the second marker. How may
                           hours have passed?
150/3 compares
quantities and is   As you can see, simple subtraction
  called rate.
                     is all that is required to answer
                               the questions.

                      The answers, 150 miles and 3
                     hours, indicate a change in miles
                    and a change in time, respectively.
                             Since each quantity describes a
                            change, the expression is a rate of
                              change. You read the rate as
                                   150 miles in 3 hours
                                             or
                                  150 miles per 3 hours.

Rates can be positive (+)
           or
                             In everyday life, most rates are
      negative (-)          given “per one unit” of a quantity.
                            This is called a unit rate. So, it is
                               more common to perform the
                             division 150/3 = 50 to arrive at a
                                    unit rate of change:
                                   50 miles per 1 hour
                                             or
                                    50 miles per hour.
After stopping to buy gas,
  a motorist drives at a
constant rate as indicated
          below.

Time (t) 0 hrs      2 hrs
Dist (d) 5 mi      120 mi

Find the speed or rate of
 change for the car then
   graph the situation.




                   Problem solving
 Calculate each rate of
         change

1) A motorist drives 140
miles in 3 hours and 280
   miles in 4.4 hours.

  2) After 5 minutes a
temperature of 35° F was
  recorded. After 6.5
minutes, the temperature
       was 32° F.

  3) A hose flows 250
   gallons of water in 2
minutes and 875 gallons in
        7 minutes.
  4) A small aircraft begins a descent to land. At 1.5
minutes the altitude is 5450 feet, and at 3 minutes the
   altitude is 4700 feet. If the plane descends at a
       constant rate, what is the rate of descent.

5) 32 people exit the stadium in 2.5 minutes and 384
      people leave the stadium in 30 minutes.

  6) A race car drives 2 miles in 0.02 hours and 305
                 miles in 3.02 hours.

    7) After 3.5 minutes of descent, a small plane’s
 altitude is 3425 feet. After 6.0 minutes, the plane’s
                   altitude if 2375.
8) Does the following data indicate a constant speed?
                        Explain.
        2 mi/4 min 8 mi/16 min 10 mi/24 min
A linear equation is an
    equation whose
solutions fall on a line
   on the coordinate
plane. All solutions of
   a particular linear
  equation fall on the
line, and all the points
     on the line are
    solutions of the
        equation.
 Look at the graph to
 the left, points (1, 3)
and (-3, -5) are found
  on the line and are
    solutions to the
        equation.
If an equation is linear, a
constant change in the x-
    value produces a
constant change in the y-
          value.

 The graph to the right
shows an example where
 each time the x-value
  increases by 2, the
y-value increases by 3.
                The equation
                  y = 2x + 6
            is a linear equation
             because it is the
            graph of a straight
           line and each time x
          increases by 1 unit, y
               increases by 2


X   Y = 2x + 6      Y      (x, y)

1    2(1) + 6        8     (1, 8)
2    2(2) + 6       10    (2, 10)
3    2(3) + 6       12    (3, 12)
4    2(4) + 6       14    (4, 14)
5    2(5) + 6       16    (5, 16)
 Using Slopes and
   Intercepts
x-intercepts and y-intercepts
x-intercept – the x-coordinate of the point
where the graph of a line crosses the x-axis
(where y = 0).
y-intercept – the y-coordinate of the point
where the graph of a line crosses the y-axis
(where x = 0).
Slope-intercept form (of an equation) – a linear
equation written in the form y = mx +b, where m
represents slope and b represents the y-
intercept.
Standard form (of an equation) – an equation
written in the form of Ax + By = C, where A, B,
and C are real numbers, and A and B are both ≠ 0.
   Standard Form of an Equation
• The standard form of
  a linear equation, you
  can use the x- and y-
  intercepts to make a
  graph.
• The x-intercept is the
  x-value of the point     Ax + By = C
  where the line
  crosses.
• The y-intercept is the
  y-value of the point
  where the line
  crosses.
 To graph a linear equation in standard form,
you fine the x-intercept by substituting 0 for
 y and solving for x. Then substitute 0 for x
                and solve for y.
 2x + 3y = 6               2x + 3y = 6
2x + 3(0) = 6             2(0) + 3y = 6
      2x = 6                     3y = 6
        x=3                       y=2

The x-intercept is 3.     The y-intercept is 2.
(y = 0)                   (x = 0)
  Find the x-intercept and y-intercept of each
line. Use the intercepts to graph the equation.

                 1)   x–y=5

                 2)   2x + 3y = 12

                 3)   4x = 12 + 3y

                 4)   2x + y = 7

                 5) 2y = 20 – 4x
Slope-intercept Form
     y = mx + b
     Slope-intercept Form

• An equation whose
  graph is a straight line
  is a linear equation.
  Since a function rule
  is an equation, a             Y = mx + b
  function can also be       (if you know the slope and where
                                  the line crosses the y-axis,
  linear.                                use this form)
• m = slope
• b = y-intercept
       For example in the equation;
                  y = 3x + 6
          m = 3, so the slope is 3
      b = +6, so the y-intercept is +6
               Let’s look at another:
                     y = 4/5x -7
           m = 4/5, so the slope is 4/5
          b = -7, so the y-intercept is -7
 Please note that in the slope-intercept formula;
                      y = mx + b
the “y” term is all by itself on the left side of the
                        equation.
              That is very important!
                      WHY?
 If the “y” is not all by itself, then we must first
 use the rules of algebra to isolate the “y” term.
           For example in the equation:

                     2y = 8x + 10

 You will notice that in order to get “y” all by itself
           we have to divide both sides by 2.
 After you have done that, the equation becomes:
                      Y = 4x + 5
Only then can we determine the slope (4), and the y-
                     intercept (+5)
       OK…getting back to the lesson…
  Your job is to write the equation of a line
after you are given the slope and y-intercept…

                   Let’s try one…

         Given “m” (the slope remember!) = 2
            And “b” (the y-intercept) = +9
     All you have to do is plug those values into
                      y = mx + b
               The equation becomes…
                      y = 2x + 9
 Let’s do a couple more to make sure you are
                expert at this.

              Given m = 2/3, b = -12,
Write the equation of a line in slope-intercept form.
                    Y = mx + b
                   Y = 2/3x – 12

                One last example…
                Given m = -5, b = -1
Write the equation of a line in slope-intercept form.
                    Y = mx + b
                    Y = -5x - 1
Writing an Equation From a Graph

 You can write an equation
from a graph. Use 2 points
to find the slope. Then use
    the slope and the y-
  intercept to write the
          equation.
 Step 1 Using the slope formula,
find the slope. Two points on the
    line are (0, 2) and (4, -1).
Step 2 write an equation in slope-
intercept form. The y-intercept
               is 2.
Write an equation of each line.




          Use points (0, 1)
         and (-2, 0)
                              Use points (0, 1)
                              and (3, -1)
Given the slope and y-intercept, write the
equation of a line in slope-intercept form.

            1) m = 3, b = -14

            2) m = -½, b = 4

            3) m = -3, b = -7

            4) m = 1/2 , b = 0

            5) m = 2, b = 4
Using slope-intercept form
    to find slopes and
       y-intercepts

   The graph at the right
  shows the equation of a
line both in standard form
 and slope-intercept form.

    You must rewrite the
   equation 6x – 3y = 12 in
 slope-intercept to be able
to identify the slope and y-
         intercept.
Using slope-intercept form to write equations,
    Rewrite the equation solving for y = to
     determine the slope and y-intercept.


    3x – y = 14                  x + 2y = 8
    -y = -3x + 14                2y = -x + 8
    -1 -1 -1                      2 2 2
    y = 3x – 14                  y = -1x + 4
    or                                2
     3x – y = 14
         3x = y + 14
    3x – 14 = y
Write each equation in slope-intercept form.
    Identify the slope and y-intercept.

                   2x + y = 10

                   -4x + y = 6

                   4x + 3y = 9

                    2x + y = 3

                     5y = 3x
Write the equation of a line in slope-intercept
 form that passes through points (3, -4) and
                   (-1, 4).

1) Find the         2) Choose either point and
   slope.              substitute. Solve for b.
4 – (-4) 8          y = mx + b (3, -4)
-1 – 3 -4           -4 = (-2)(3) + b
m = -2              -4 = -6 + b
                    2=b
                    Substitute m and b in equation.
                    Y = mx + b
                    Y = -2x + 2
     Write the equation of the line in
slope-intercept form that passes through
            each pair of points.
           1)   (-1, -6) and (2, 6)

           2)   (0, 5) and (3, 1)

           3)   (3, 5) and (6, 6)

           4)   (0, -7) and (4, 25)

           5)   (-1, 1) and (3, -3)
Graphing an Equation
     y = 3x -1                  The slope is 3,
                                use the slope to
                                plot the second
                                point


                                                       
                                                   
              The y-intercept
              is -1, so plot
              point (0, -1)



          



                                      Draw a line
                                      through the
                                      two points.
   Point-Slope Form
Writing an equation when you know
a point (2, 5) and the slope m = 2
 In the graph below, use the information
provided to write the equation of the line.
   Use what you know about writing an
     equation in slope-intercept form.




                                 Slope = 2 and
                                 point (2,7)
 Do you think you can use the same method to
    find the y-intercept in the graph below?
 Here we must use a different form of writing
an equation and that form is called point-slope.




                                  Slope = 7/3
                                  and point (2,7)
• Suppose you know that a
  line passes through the
                                 Point-Slope Form
  point (3, 4) with slope 2.        and Writing
  You can quickly write an           Equations
  equation of the line using
  the x- and y-coordinates
  of the point and using the
  slope.                       y – y1 = m(x – x1)
                                (if you know a point and the
• The point-slope form of             slope, use this form)
  the equation of a
  nonvertical line that
  passes through the
  (x1, y1) with slope m.
       Let’s try a couple.



Using point-slope form, write the equation of a line
       that passes through (4, 1) with slope -2.
                  y – y1 = m(x – x1)
                 y – 1 = -2(x – 4)Substitute 4 for x , 1 for y and -2 for m.
                                                           1     1




              Write in slope-intercept form.
                      y – 1 = -2x + 8
                        y = -2x + 9
     One last example

Using point-slope form, write the equation of a line
       that passes through (-1, 3) with slope 7.
                 y – y1 = m(x – x1)
                 y – 3 = 7[x – (-1)]
                  y – 3 = 7(x + 1)

          Write in slope-intercept form
                   y – 3 = 7x + 7
                    y = 7x + 10
  If you know two points on a line, first use
them to find the slope. Then you can write an
         equation using either point.
• Step one – Find the
  slope of a line with
  points (-4, 3), (-2, 1)   y2  y1
                                    m
                            x2  x1
                               1 3      2
                                            1
                             2   4 2
Step Two – Use either point to write the
equation in point-slope form. Use (-4, 3)


               y – y1 = m(x – x1)
              Y – 3 = -1[x – (-4)]
               Y – 3 = -1(x + 4)

        Write in slope-intercept form
               Y – 3 = -1(x + 4)
                 Y – 3 = -x - 4
                   Y = -x - 1
Writing Equations of Parallel
 and Perpendicular Lines
        Geometry Connection
    In coordinate geometry you studied how to
        determine if lines where parallel or
                  perpendicular.



• Nonvertical lines are parallel if they have the same
  slope and different y-intercept. Any 2 vertical lines
  are parallel. (y = 3x + 1 and y = 3x -3)

• Two lines are perpendicular if the product of their
  slopes is -1. A vertical line and a horizontal line are
  also perpendicular. (y = -¼x 1 and y = 4x + 2)
In the graph on the left, the two lines are parallel. Parallel
       lines have the same slope and never intersect.
 In the graph at the right, the two lines are perpendicular.
 Perpendicular lines are lines that intersect to form right
                           angles.
    Writing Equations of
       Parallel Lines

    Write the equation for the line that contains (5, 1) and is
                       parallel to y = ¼x – 4.

Identify the slope of the given line. y = ¼x – 4 (slope is ¼)
Using point-slope form, write the equation.
y – y1 = m(x – x1)      point-slope form of an equation

y – 1 = ¼(x – 5)        substitute (5, 1) for (x1, y1) and ¼ for m

Y – 1 = ¼x – 5/4        using the Distributive Property, remove the parentheses

Y = ¼x – ¼              simplify and rewrite in slope-intercept

                 The equation is y = ¼x – 1/4 .
1) Write an equation for the line that contains
     (2, -6) and is parallel to y = 2x + 9.




    2) Write an equation for the line that
  contains (3, 4) and is parallel to y = ½x – 4.
  For perpendicular lines, the product of two
 numbers is -1, if one number is the negative
 reciprocal of the other. Here is how to find
     the negative reciprocal of a number.
Start with a fraction: 3     Start with an integer: 4
                         5
Find its reciprocal: 5       Find its reciprocal:   1
                    3                               4

Write the negative 5         Write the negative 1
                                              
  reciprocal:       3          reciprocal:       4
  Writing Equations of
  Perpendicular Lines

Write an equation of the line that contains (0, -2) and is
  perpendicular to y = 4x + 3.

Identify the slope of the given line. Y = 4x + 3 (slope is 4, the
    negative reciprocal is –¼).
Using point-slope form, write the equation.
y – y1 = m(x – x1)       point-slope form if an equation.

y –(-2) = -¼(x – 0)      substitute (0, -2) for (x1, y1) and –¼ for m

 y + 2 = -¼x – 0         using the Distributive Property, remove the parentheses

Y = -¼x - 2              simplify and rewrite in slope-intercept.

                 The equation is y = -¼x – 2.
1) Write an equation of the line that contains (1, 8)
        and is perpendicular to y = ¾x + 1.


2) Write an equation of the line that contains (6, 2)
        and is perpendicular to y = -2x + 7
Graphing Absolute
 Value Equations
  Distance from zero?
    A V-shaped graph that points upward or
   downward is the graph of an absolute value
                   equation
• The Absolute value of a
  number is its distance from 0
  on a number line.
• Make a take of values and
  graph the equation y = |x| + 1
Below are the graphs of y = |x| + 1 and y = |x| + 2.
 Describe how the graphs are the same and how
               they are different.



            y = |x| + 1
                                          y = |x| + 2




The graphs are the same shape. The y-intercept of
  the first graph is 1. the y-intercept of the second
  graph is 2
Describe how each graph below is like y = |x|
          and how it is different.




               y = |x| + 3



                                          y = |x| - 3
   Graph each equation
       (function)


1) y = |x| + 2

2) y = |x| – 4

3) y = |x| + 1

4) y = |x| - 5
Use x-values of -2, -1, 0, 1, 2
     Equation Forms
        (review)
When working with straight lines, there
 are often many ways to arrive at an
         equation or a graph.
 Slope Intercept Form

If you know the slope and where the line crosses the
                 y-axis, use this form.


                     y = mx + b

                       m = slope
                    b = y-intercept
             (where the line crosses the y-axis)
  Point Slope Form

If you know a point and the slope, use this form.


             y – y1 = m(x – x1)

                   m = slope

          (x1, y1) = a point on the line
    Horizontal Lines
              y=3     (or any number)
Lines that are horizontal have a slope of zero. They
   have “run” but no “rise”. The rise/run formula for
         slope always equals zero since rise = o.
                      y = mx + b
                      y = 0x + 3
                         y=3
  This equation also describes what is happening to
    the y-coordinates on the line. In this case, they
                      are always 3.
             Vertical Lines

                      x = -2
       Lines that are vertical have no slope
                 (it does not exist).
They have “rise”, but no “run”. The rise/run formula
   for slope always has a zero denominator and is
                       undefined.
 These lines are described by what is happening to
    their x-coordinates. In this example, the x-
          coordinates are always equal to -2.
There are several ways to graph a straight line
             given its equation.
  Let’s quickly refresh our memories on equations of straight
                             lines:



 Slope-intercept            Point-slope        Horizontal line            Vertical line
    y = mx + b           y - y1 = m(x – x1)    Y = 3 (or any #)         X = -2 (or any #)
 When stated in “y=”     when graphing, put    Horizontal lines have    Vertical line have no
form, it quickly gives    this equation into     a slope of zero –        slope (it does not
  the slope, m, and      “y=” form to easily   they have “run”, but      exist) – they have
   where the line           read graphing      no “rise” – all of the   “rise”, but no “run” –
crosses the y-axis, b,       information.         y values are 3.        all of the x values
    called the y-                                                               are -2.
     intercept.
                 Remember

                              Before graphing a line,
If a point lies on a line,        be sure that your
  its coordinates make          equation starts with
    the equation true.                   “y =”
    (2, 1) on the line         To graph 6x + 2y = 8
   y = 2x -3 because            rewrite the equation:
       1 = 2(2) - 3                 2y = -6x + 8
                                    Y = -3x + 4
                             Now graph the line using
                               either slope intercept
                                   method or table
                                       method.
Practice with Equations
        of Lines
   Writing and graphing lines
     Practice with Equations of Lines
 Answer the following questions dealing with equations
             and graphs of straight lines.




1)   Which of the following equations passes
     through the points (2, 1) and (5, -2)?

a.   y = 3/7x + 5             b. y = -x + 3
c.   y = -x + 2               d. y = -1/3x + 3
2) Does the graph of the straight line
 with slope of 2 and y-intercept of 3
   pass through the point (5, 13)?


                               Yes

                               No
3) The slope of this line is 3/2?


    True



    False
     4) What is the slope of the line
             3x + 2y = 12?
a)   3
b)   3/2
c)   -3/2
d)   2
5) Which is the slope of the line
   through (-2, 3) and (4, -5)?


                      a)   -4/3
                      b)   -3/4
                      c)   4/3
                      d)   -1/3
6) What is the slope of the line shown in the
               chart below?


    X       1       3          5         7
    Y       2       5          8         11




                          a)       1
                          b)       3/2
                          c)       3
                          d)       3/5
7) Does the line 2y + x = 7 pass
   through the point (1, 3)?


                            True

                           False
     8) Which is the equation of a line
        whose slope is undefined?


a)   x = -5
b)   y=7
c)   x=y
d)   x+y=0
9) Which is the equation of a line that
passes through (2, 5) and has slope -3?

              a)   y = -3x – 3
              b)   y = -3x + 17
              c)   y = -3x + 11
              d)   y = -3x + 5
10) Which of these equations
represents a line parallel to the line
            2x + y = 6?



                      a)   Y = 2x + 3
                      b)   Y – 2x = 4
                      c)   2x – y = 8
                      d)   Y = -2x + 1

								
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