Multiple Unknowns

In order to find a numerical solution to an equation, the equation must contain only one variable. That means that there is only one unknown value.  Most word problems, however,  have more than one unknown. For example, you may want to know the dimensions of a rectangle; both the length and the width. You may wish to determine how much money to invest in two separate accounts in order to earn a particular amount of interest. No matter what the situation, you will need to express all unknowns in terms of one variable. In order to do this, the unknown quantities must be somehow related. You will need to determine how they are related and use that relationship to express all the unknowns in terms of one variable. One of the unknowns will always be the variable. For our purposes here in this tutorial,  we will let x be that variable. 

 Identifying the variable 

There are a couple of things to consider when trying to identify the quantity that  the variable should represent. The best place begin is to look at  the problem question. If you are looking for the width of a rectangle, then you may wish to let x be the width. Further investigation into the problem statement will help you to confirm this choice. Sometimes the problem is not specific. For example, instead of a question, the problem may simply sate: "Find the dimensions of the rectangle." In such cases, you may have to look elsewhere in the problem. 

When investigating the problem statement to either confirm your choice for the variable assignment or determine  the variable assignment, look for a relationship between the unknown quantities. For example, if both the length and the width are unknown, then there should be a statement relating these two quantities such as: "the length is twice the width."  With this type of comparative statement, it is best to let the variable represent the quantity being compared to. In this case, the length is being compared to the width so let x represent the width. In a direct comparison statement, the subject of the sentence is compared to the object. The object is found at the end of the sentence. Let x be the value that is represented at the end of the sentence. 

 Expressing the other unknowns in terms of x 

The remainder of the sentence can then be translated directly to express the other unknown in terms of x.  For example: 

The length is three more than the width.

The length is being compared to the width. 

Let x = width

The length is three more than the width.

The length =                   3  +              x   

width = x
length = x + 3

Sometimes the relationship is indirect. You will need to understand terms. 

Example 1:  "Find the radius of a circle whose diameter is 10 cm." 
                      Here the relationship between radius and diameter is implied by geometry.
                      The radius is half the diameter therefore the radius is 5 cm. 

Example 2: "A sweater was discounted 20%."
                      Discounts are percentages of the original price. If x is the original price
                      of the sweater, then the discount can be expressed as 0.20x.

 more examples

The following are comparative statements relate two unknown quantities. Based on the statement, represent the two numbers in mathematical terms:

 

  • The greater of two numbers is 8 less than 4 times the smaller number.

 The greater number is being compared to the smaller number. 
                                                                                        
                                                                                          x

                                                                                          x
                                                                                        
The greater of two numbers is 8 less than 4 times the smaller number.
                                                                          
         greater number             =          4 x - 8

Let X = the smaller number
Solution                                   
The smaller number:         x    
The greater number:       4x - 8

 
 

  • The second of two numbers is 7 more than 4 times the first.

 Solution                                   
The first number:         x    
The second number:  4x + 7
 		 
 
  • The width of a rectangle  is 5 less than half the length. 
 Solution                                   
The length:         x    
The width:       ½  x - 5
 		 
 
  • The second of two numbers is  8 less than twice the first.
 Solution                                   
The first number:         x    
The second number:  2x - 8     
 		 
 
  • the greater of two numbers is 6 more than 3 times the smaller.
 Solution                                   
The smaller number:         x    
The greater number:       3x + 6
 		 
 
  • One number is 21 less than twice the other.
 Solution                                   
The other number:         x    
One number:            2x - 21
 		 
 
  • Three consecutive integers.
 There is an implied relationship here. Consecutive integers increase by one. In other words, the second is one more than the first. [for example: 1,2,3,4,5,6,7,8...x, x+1, x+2, ...] The third is one more than the second, and so on.  Let X be the first integer

Solution                                   
The first  integer:         x    
The second integer:    x + 1
The third integer:         x + 2

 
 
  • Two consecutive, odd integers
 Even and odd consecutive integers increment by two. 


Even integers:  ... -2, 0, 2,4,6,8,10,...x, x+2, x+4,...
Odd integers:     ... -3,-1,1,3,5,7,9,....x, x+2, x+4,...

 In other words, the second is two more than the first. The third is two more than the second, and so on.  Let X be the first integer.

Solution                                   
The first odd  integer:         x    
The second odd  integer:    x + 2
The third odd  integer:         x + 4

 
 
  • Two consecutive, even integers
 Solution                                   
The first even integer:         x    
The second even integer:    x + 2

NOTE: the expressions for even and odd integers look the same. The difference is in assigning the initial x to an even integer versus an odd integer. 

 
 

 

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