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 = c
where a, b, and c are constants.
3x - 4 = 5
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 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 that it doesn’t matter what side the variable is on. -6 = a means the same thing as a = -6.
However, most times, we have to use several properties to get the job done. The following is a strategy that you can use to help you solve linear equations that are a little bit more involved.
Step 1: Simplify each side, if needed.
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.
What it boils down to is that
you want to get
the variable you are solving for alone on one side and everything else
on the other side using INVERSE operations.
Example 4: Solve the equation .
*Inverse of mult. by -3 is div. by -3
If you put 1 back in for x in the original problem you will see that 1 is the solution we are looking for.
*Inverse of sub. 5 is add 5
*Inverse of mult. by -1 is div. by -1
*Inverse of sub. 2 is add 2
*Inverse of mult. by 7 is div. by 7
If we let x represent the first integer, how would we represent the second consecutive integer in terms of x? Well if we look at 5, 6, and 7 - note that 6 is one more than 5, the first integer.
In general, we could represent the second consecutive integer by x + 1. And what about the third consecutive integer.
Well, note how 7 is 2 more than 5. In general, we could represent the third consecutive integer as x + 2.
Consecutive EVEN integers are even integers that
follow one another
in order.
If we let x represent the first EVEN integer, how would we represent the second consecutive even integer in terms of x? Note that 6 is two more than 4, the first even integer.
In general, we could represent the second consecutive EVEN integer by x + 2.
And what about the third consecutive even integer? Well, note how 8 is 4 more than 4. In general, we could represent the third consecutive EVEN integer as x + 4.
Consecutive ODD integers are odd integers that
follow one another
in order.
If we let x represent the first ODD integer, how would we represent the second consecutive odd integer in terms of x? Note that 7 is two more than 5, the first odd integer.
In general, we could represent the second consecutive ODD integer by x + 2.
And what about the third consecutive odd
integer? Well, note how
9 is 4 more than 5. In general, we could represent the third
consecutive
ODD integer as x + 4.
Note that a common misconception is that because we want an odd number that we should not be adding a 2 which is an even number. Keep in mind that x is representing an ODD number and that the next odd number is 2 away, just like 7 is 2 away form 5, so we need to add 2 to the first odd number to get to the second consecutive odd number.
If x represents the first of four consecutive integers, express the sum of the four integers in terms of x.
We can represent them the following way:
x + 1 = 2nd consecutive integer
x + 2 = 3rd consecutive integer
x + 3 = 4th consecutive integer
Second we need to write it as a sum of the four integers and then simplify it:
If x represents the first of three odd consecutive integers, express the sum of the first and third integers in terms of x.
We can represent them the following way:
x + 2 = 2nd consecutive odd integer
x + 4 = 3rd consecutive odd integer
Second we need to write it as a sum of the first and third odd integers in terms of x and then simplify it:
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 - 1d: Solve the given equation.
Practice Problem 2a: Write an algebraic expression and simplify if possible.
Need Extra Help on these Topics?
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 26, 2011 by Kim Seward.
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