Beginning Algebra Tutorial 13

Beginning Algebra
Tutorial 13: Multiplication Property of Equality

WTAMU > Virtual Math Lab > Beginning Algebra

After completing this tutorial, you should be able to:

1. Use the multiplication and division properties of equalities to solve linear equations. 
2. Solve an equation using more than one property.
3. Know how to express consecutive integers in terms of x, if the first integer is x.
4. Know how to express even consecutive integers in terms of x, if the first even integer is x.
5. Know how to express odd consecutive integers in terms of x, if the first odd integer is x.

As mentioned in Tutorial 12: Addition Property of Equality, solving equations is getting into the heart of what algebra is about.  As we did in Tutorial 12,  we will be looking specifically at linear equations and their solutions.  We will start off slow and solve equations that use only the multiplication or division property of equality to make sure you have the individual concepts down.  Then we will pick up the pace and mix 'em up where you need to use several properties and steps to get the job done. 

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.

 
 
  Equation

Two expressions set equal to each other.

 
  Linear Equation 
in One Variable

An equation that can be written in the form 
ax + b = c
where a, b, and c are constants.

 
 

3x - 4 = 5

 
  Solution

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)

 
 
  Solution Set

Set of all solutions

 
  Solving a Linear Equation 
in General

Get the variable you are solving for alone on one side and everything else on the other side using INVERSE operations.

 
 
 
  Multiplication and Division Properties of Equality

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.

 
  Strategy for Solving a Linear Equation
 

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

 

 
 
 
  Consecutive Integers
 

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:

 
 
 

These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

1a. 
(answer/discussion to 1a)

1b. 
(answer/discussion to 1b)

1c. 
(answer/discussion to 1c)

1d. 
(answer/discussion to 1d)

2a.  If x represents the first of three consecutive integers, express the sum of the three integers in terms of x.
(answer/discussion to 2a)

WTAMU > Virtual Math Lab > Beginning Algebra