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Linear Equations


Read this before you try to do the problems! :)
 An EQUATION  is a statement in which two expressions are equal. A LINEAR EQUATION is one variable is an equation that can be written in the form of ax = b where a and b are constants and a never equals zero. A number is a SOLUTION of an equation if the statement is true when the number is substituted for the variable.
     Two equations are EQUIVALENT if they have the same solutions.  For instance, the equations x-4 = 1 and x = 5  are equivalent because both have the number 5 as their only solution. The following transformations, or changes, produce equivalent equations and can be used to solve an equation.

-Addition Property of Equality Add the same number to both sides: If a = b, then a+c = b+c.

-Subtraction Property of Equality= Subtract the same number from both sides: If a = b, the a-c = b-c.

-Multiplication Property of Equality= Multiply both sides by the same nonzero number: If a = b and c is not zero, then ac = bc.

-Division Property of Equality= Divide both sides by the same nonzeronumber: If a = b and c is not zero, the a/c = b/c.

***Your goal is to isolate the variable on one side of the equation!
***Always be sure to check your solution to each answer!
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(-5)x = 10-15
(-5)x = -5
(-5)x = -5
 -5        -5
x = -1
 

3-2x = 4
(3-2x)-3 = 4-(3)
-2x = 1
-2     -2
x = -1/2
 

6z+5z-3 = 41
11z-3 = 41
(11z-3)+3 = (41)+3
11z = 44
 11     11
z = 4
 

Literal Equations


Some things you should know before you begin...
With literal equations you can sole an equation that involves more than one variable for one of its variables.  This is simple because you use all the pre mentioned properties to put all but one variable on one side of the equation.  The easiest way to explain this is to guide you through some very simplistic examples. :)
T = Q/V   for V
V = Q/T
 

W = Fd   for d
d= W/F
 
 

4x+2y = 8 for x
(4x+2y)-2y = (8)-2y
4x = -2y+8
 4        4
x = (-1/2)y+2
 
 

K = 2mgh   for h
h = (1/2)K/mg

 
***If you wanted to find V it would have to be on its own side of the equal sign.  As you can tell, it now shares a side with the Q.  By multiplying both sides by V you get the equation, TV = Q.  Now the V shares a side with the T.  To get rid of the T you just divide both sides by T.  You are now left with the equation of V = Q/T which is the answer to the question because you now have found out what V is in relation to Q and T. 

***The question asks you to find the value of the variable d.  As you can see d is sharing a side of the equation with F, so to get rid of F you divide both sides by F.  The resulting equation is, d = W/F.  This is the answer to the question asked because you now have d on one side of the equation all alone.

***You start out with an equation with two variables on the same side preceded by constants that is equal to the constant number 8.  To get the x you would first get rid of the y's by subtraction (because they are added to the x's).  You then are left with a constant in front of the x and this is not acceptable!  Since x is being multiplied by 4 you would divide both sides of the equation by 4.  You now have x = (-2/4)y+(8/4).  You should always simplify your answers, and you would end up with the resulting equation of x = (-1/2)y+2.

***Uh oh!  Now we have added in a constant with the variables!  But no sweat, they can be put onto the other side of the equation just as easily as a letter variable.  You want to get rid of the 2 first, so you would divide both sides by 2.  The resulting equation is: K/2 = mgh.  (As we know, K/2 is the same thing as (1/2)K.  Now you can divide both sides by m.  This equation is: (1/2)K/m = gh  Now you would divide both sides by g.  The resulting, and final equation is: (1/2)K/mg = h.  (***You could do this much faster by dividing the original equation by 2mgh.  Your result still would have been (1/2)K/mg = h.  It is O.K. to break off this one portion of the equation because it is all multiplied eventually and will end up being divided eventually :)***)

 
 

Getting the Equation 
For a Line


General Information before you begin...



 

 Slope intercept- Given the slope m and the y-intercept b, use this equation:
y = mx+b

Point-slope- Given the slope m and a point (x1,y1), use this equation:
y-y1 = m(x-x1)

Standard- Given two points (x1,y1) and (x2,y2) use the formula
ax+by = c
In order to find the slope of a line you can use the following formula when given two points:
m = y2-y1
      x2-x1

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 A line with the slope of (3/2) through the point (0,-1)
***What you given here is the slope and a point at which it passes through.  Using this information you can first find the point slope equation, which is (using the pre mentioned formula) y+1 = (3/2)(x-0).  Once you have this you can use what you learned in solving linear and literal equations to solve the formulas for slope intercept and standard form. 
Slope intercept ->   y = (3/2)x-1
Point-slope ->   y+1 = (3/2)(x-0)
Standard ->   3x-2y = 1
 
 
 

A line with the slope of 7 through the point (-5,3)
Slope intercept ->   y = 7x+8
Point-slope ->   y-3 = 7x+5
Standard ->   y+x = -8/7
 
 
 

Write an equation of the line that passes through (1,-1) and is perpendicular to the line y = (-1/2)x+6
***In this problem you are given the point of the line in question and the equation for a line that is perpendicular to it.  If you know what perpendicular means, then you know that the slope of a line perpendicular

 
 

This page constructed by: Heather S.
October 5, 2001 for Mrs. Felz Honors Algebra II class, 3rd period
 
 

my LInks:

 To get to my class mates pages click here for the gallery!
 
This is from her second period classes.

 This is my school's (Union Grove High School) web site, it is pretty nifty.