Learning Objectives
Introduction
Equations can be used to help us solve a variety of problems. In later tutorials, we will put them to use to solve word problems.
Tutorial
Two expressions set equal to each other
An equation that can be written in the form
ax + b = 0
where a and b are constants
A value, such that, when you replace the variable with it, it makes the equation true.
(the left side comes out equal to the right side)
Set of all solutions
Get the variable you are solving for alone on one side
and
everything
else on the other side using INVERSE operations.
If a = b, then a + c = b + c
If a = b, then a - c = b - c
Note that addition and subtraction are inverse
operations of each
other. For example, if you have a number that is being added that
you need to move to the other side of the equation, then you would
subtract
it from both sides of that equation.
Example 1: Solve for x.
Example 2: Solve for x.
If a = b, then a(c) = b(c)
If a = b, then a/c = b/c where c is not equal to 0.
Note that multiplication and division are inverse operations of each other. For example, if you have a number that is being multiplied that you need to move to the other side of the equation, then you would divide it from both sides of that equation.
Note, for multiplication and division, it is not
guaranteed that if
you multiply by the variable you are solving for that the two sides are
going to be equal. But it is guaranteed that the two sides are
going
to be equal if you are multiplying or dividing by a constant or another
variable that you are not solving for. We will talk more about
this
in a later tutorial. For this tutorial just note you can use this
property with constants and variables you are not solving for.
Example 3: Solve for x.
Example 4: Solve for x.
Step 1: Simplify each
side, if needed.
To remove ( ): Just use the distributive property.
To remove fractions: Since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.
Step 2: Use Add./Sub. Properties to move the variable term to one side and all other terms to the other side.
Step 3: Use Mult./Div. Properties to remove any values that are in front of the variable.
Step 4: Check your
answer.
I find this is the quickest and
easiest way
to approach linear equations.
Example 5: Solve for y.
*Inverse of mult. by -4 is div.
by -4
If you put -1 back in for y in the original problem you will see that -1 is the solution we are looking for.
Example 6: Solve for x.
*Get all the x terms on one side
*Inverse of add. 150 is sub. 150
*Inverse of mult. by 7 is div. by 7
Example 7: Solve for x.
*Get all x terms on one side
*Inverse of sub. 2 is add. 2
*Inverse of mult. by -5 is div.
by -5
An equation is classified as an identity when it is true
for
ALL real numbers for which both sides of the equation are defined.
Example 8: Solve for x.
*Get all the x terms on one side
Whenever your variable drops
out AND you end
up with a TRUE statement, then the solution is ALL REAL NUMBERS. This
means that if you plug in any real number for x in this equation, the left side will equal the right side.
Also note that in line 2 above,
both sides of the equation have the same expression. This is
another sign that this equation is an identity.
So the answer is all real numbers, which means this equation is an identity.
A conditional equation is an equation that is not an
identity,
but has
at least one real number solution.
Example 9: Solve for x.
This would be an example of a conditional equation, because we came up with one solution.
An inconsistent equation is an equation with one variable that has no solution.
Example 10: Solve for x.
*Get all the x terms on one side
Whenever your variable drops out AND you end up with a FALSE statement, then after all of your hard work, there is NO SOLUTION.
So, the answer is no solution which means this is an inconsistent equation.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: Solve the given equation.
Practice Problems 2a - 2c: Determine whether the equation is an identity, a conditional equation or an inconsistent equation.
Need Extra Help on these Topics?
http://www.purplemath.com/modules/solvelin.htm
This webpage helps you to solve linear equations.
http://www.mathpower.com/tut50.htm
This webpage gives an example of solving a linear equation.
http://www.sosmath.com/algebra/solve/solve0/solve0.html#linear
This website helps you to solve linear equations. Only
do the linear equations at this site. Do not go on to equations
containing
radicals - that will be covered in a later tutorial.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 16, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward.
All rights reserved.