SWOPE'S SLOPES
  Linear EquAtiOns
Linear Equations are equations that when solved, 1 variable equals 1 number.

 
5y - 6 = 2 - 4y
7     7		 Given problem.
7 (5y) - 7 (6) = 7 (2 - 4y)
1   7      1  7      1		 You want to get rid of the fractions, so you 
multiply both sides by 7 to
                                   1
cancel out the 7 in the denominator.
5y - 6 = 14 - 28y Your new problem after multiplying by 7
                                                              1

     5y - 6 = 14 - 28y
 +28y+6   +6 +28y		 Now you can start to actually work the 
problem. You need all alike variables on 
one side.
step one:  add 28y to both sides
step two:  add 6 to both sides
33y = 20 Your new problem after combining like terms.
33y = 20
33      33		 Now you need to isolate the 'y'. All you have 
to do is divide by 33 .
y = 20
      33		 Your answer, y = 20
                            33
 
  LITERAL EQUATIONS


Literal Equations have more than 1 variable.  They are solved for a specific variable.
      A numerical answer is not available for this type of equation.


 
So = Do  : Di
Si     Di		 Given problem. The ':' means to find for, so you are solving for Di.
Di ( So ) =  ( Do ) Di
1     Si           Di     1		 First, multiply both sides by Di.
                                             1
This gets Di on the left, and cancels it on the right.
Di (So) = Do
      Si		 Your new problem.
Di (Si )(So ) = Do (Si )
      So  Si               So		 Now you have to get Di by itself. To do this, multiply both sides by Si.
                                           So
Di = DoSi
         So		 Your answer, Di = DoSi
                                 So

 

WRITING LINEAR EQUATIONS

---Slope Intercept Form---

Slope Intercept Form is y=mx+b, where m is the slope and b is the y-axis intercept.

 
A line goes through point (5,2) and its slope is -3
                          5		 Given problem. Put into y=mx+b form.
2 = -3 ( 5) + b
       5		 Put the points into the 'x'(5) and 'y'( 2) spots. Put the -3 in as the 'm'
             5
2 = -3 (5) + b
       5		 The 5 in the denominator cancels the 5 in the numerator. Leaving you with -3.
  2 = -3 + b
+3  +3		 Your new problem. Now you are trying to isolate 'b'. Add 3 to both sides , and that will give you what b equals.
5 = b B equals 5.
y = -3x + 5
       5		 Go back to y=mx+b and plug in your 'b' and 'm' values.
y = -3 x + 5
       5		 Your answer, y = -3 x + 5
                             5

, THERE'S ANOTHER WAY!

~ ~ ~Point Slope Form~ ~ ~

  Point Slope Form is Y - Y1 = M (X - X 1), where X1 and Y1 are the coordinates of a point on the line and M is the slope.

A line goes through point (0,-1) and has a slope of 3.
                               2		 Given problem. You are wanting to put this info into Y- Y1 = M (X-X1) form.
Y + 1 = 3 (X - 0)
             2		 Substitute in your 'x' and ' y' values (given as the coordinates from the point) and also your 'm' value, or slope. Remember that the negatives turn into a positive (with the 1).
Y + 1 = 3 (X - 0)
             2		 Your new problem. Oddly enough, this is also your answer.
Point Slope Form is very quick and easy. It also can go into y=mx+b with a few more steps.
                    
Y + 1 = 3x
              2		 First distribute the  3 to the X and 0.
                              2
Y +1 = 3x
    -1     2  -1		 Now just subtract the 1 from both sides .
Y = 3 x - 1
       2		 This leaves you with your y=mx+b answer, which is, y = 3 x-1
                                  2

 
 
 

* * * Standard Form* * *

Standard Form is Ax + By = C, where A is the 'x' value of a point, B is the 'y' value
and C is the y intercept. All the variables are on one side.



A line goes through point (2,3) and has a slope of -1
                                2		 Given problem. Standard Form is mainly used when the equation is already in Point Slope or Slope Intercept form.
3 = -1(2) + b
       2		 First put it into y=mx+b form, and solve for 'b'.
3 = -1(2) + b
       2		 The 2's cancel, leaving you with a new problem.
   3 = -1 + b
  +1   +1
  4 = b		 Your new problem. Now, you add one to both sides and that will solve for b. B equals 4.
     y = -1x + 4
            2
+ 1x   + 1x
   2        2		 Go back to y=mx+b and plug in all your values. Now we start putting it into Standard Form. Add 1x to both sides.                         2
(2) 1x + ( 2)y = 4(2)
     2		 Your new problem. Now the variables are on one side. BUT in Standard Form there are NO FRACTIONS. All you have to do is multiply both sides by 2 . 
x + 2y = 8 Your answer, x + 2y = 8
Hint: If the problem is given in Standard Form and is wanting you to put it into Slope Intercept or Point Slope Form, there is a quick and easy formula to help you find the slope and y intercept. Slope = A  and Y inter. = C                                  -B
                       B

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