D = rt is a formula for distance traveled by an object going at a

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					               Class Notes on Unit AE: Solving Literal Equations

D = rt is a formula for distance traveled by an object going at a rate of r for a time of t.
          If r is 60 miles per hour and time is 3 hours then distance is 60(3) or 180 miles.
          If the distance traveled was 300 miles and the time was 6 hours one could
         solve the equation 300 = r(6). The solution is 50 miles per hour.
         One could also take the formula D = rt and solve for r. Then you would have a
         formula for rate.
           D = rt            To “solve for r”, you must isolate r on one side of the equation.
           D   rt          Divide each side of the equation by t. Because we
             =             do not know the value of D and t, we cannot actually
           t    t
                           do the division. Instead we just show D divided by t.

           D                                                         D
             = r           Now we have a formula for Rate. r =         or Rate is equal to
           t                                                         t
                           Distance divided by Time.


y = mx + b is called the slope-intercept formula for the graph of a line. m is the line’s
       slope. (0,b) is the line’s y-intercept. (x,y) represents the coordinates of any
       point on the line. If a line has a slope of –3 and (0,9) is it’s y-intercept, the
       equation of the line in slope-intercept format is y = -3x + 9.
         Suppose that we wanted a formula for the slope of the line. One could solve for
         m as follows:
         y = mx + b              To “solve for m”, you must isolate m on one side of the equation.
           y – b = mx        Begin by subtracting b on both sides of the equation.
         y  b
               =m            Now divide each side of the equation by x.
           x
               y  b         This is a formula for the slope of a line where (x,y) is any point on
         m=                  the line where the x-coordinate is not zero, and b is the y-
                 x
                             coordinate of the y-intercept.
Try theses1:

                a.    a + b = c             Solve for b.                   b.   q = pt – s           Solve for t.




1                                 q  s
    a.   b=c–a        b.   t =
                                    p
                                                                                                            AE-2
Suppose (1,3) is a point on a line and (0,5) is the line’s y-intercept.
          Find the slope of this line                           Write the equation of the line in
                                                                     slope-intercept form.
           y  b           Take the formula for
m=                         slope.                            y = mx + b           Take the slope-intercept
             x
                                                                                  formula for a line.
           3  5           Substitute the values                                  Substitute the values of
m=                         given for x, y and b.             y = -2x + 5          m and b. Note that (x,y)
             1
                           Evaluate.                                              represents any point on
m = -2                     The slope is -2                                        the line.


In order to sketch a graph using a graphing calculator or using certain by hand methods,
one has to “solve for y.” Here is an example with a linear equation.

      2x + 3y = 12                          The goal is to put the equation in the form y = mx + b.
               3y = -2x + 12                First subtract 2x on both sides of the equation.
               3y           2x  12
                    =                       Divide both sides of the equation by 3.
               3               3
          3y             2x   12
                 =         +               Make into two fractions (reverse of addition)
          3              3     3
                               2
                y   =      -     x + 4      Reduce constant and separate coefficient from the variable x.
                               3


c.             Solve2 for y:      -6x + 4y = -28              c.       Solve3 for y:        -3x - 4y = -12




2
     c.    y= 3x     - 7
             2
3
     c.    y=-3x        + 3
              4

				
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