# 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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