Math Lesson 8.1.4 - Substituting into a Formula

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Welcome to our Math lesson on Substituting into a Formula, this is the fourth lesson of our suite of math lessons covering the topic of Writing Formulas and Substituting in a Formula, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Substituting into a Formula

The next step after writing (correctly) a formula is to substitute the known variables in it. In this way, we open the path to the calculation of the unknown quantity. For example, in the previous exercise we can substitute h = 20 and calculate how much money Matt will have in total after 20 hours of work. Thus, since the general rule applied in this situation corresponds to the formula we wrote at the end of the solved example,

where M(h) represents the amount of money Matt will have after working for h hours, substituting h = 20 yields

Example 4

The first five numbers (terms) in a sequence are 8, 11, 14, 17 and 20.

1. Write down a formula that allows us find the n-th term of the sequence in terms of the first term without having need to work them out one by one.
2. Use this formula to find the 23th term of the given sequence.

Solution 4

1. From the first terms of the sequence it is obvious that they increase by 3, which means the difference between the successive and the previous term is always 3. Therefore, if we denote a whatever term of this sequence by an, then the previous term is the a_{n - 1}-th term. Given that the difference is d = 3, we can write
   a_n - a_{n - 1} = d
   or
   a_n - a_{n - 1} = 3
   Given that
   a₂ = a₁ + d
   a₃ = a₂ + d = (a₁ + d) + d = a₁ + 2d
   and so on, we obtain for the n-th term in terms of the first one:
   a_n = a₁ + (n - 1)d
   This is the formula for the n-th term of an arithmetic sequence, for which we will discuss more extensively in chapter 12.
   Substituting a₁ = 8 and d = 3 yields for this specific case
   a_n = 8 + 3(n-1)
   = 8 + 3n - 3
   = 3n - 5
2. Given the above formula, we obtain for the 23th term of this sequence (n = 23)
   a_n = 3n - 5
   a₂3 = 3 ∙ 23 - 5
   = 69 - 5
   = 64

From the above example, it is easy to identify the steps one has to follow when dealing with formulas:

Step 1: Identify the variables that are present in the situation under consideration.

Step 2: Assign a different letter to every variable including the one you are going to calculate.

Step 3: Write down the correct formula by including all variables identified in the first step.

Step 4: Substitute the other variables with numbers and calculate the value of the variable you are interested in by using the BEDMAS (PEMDAS) rule.

Example 5

The formula of position x in respect to the time elapsed t for an object moving in a uniformly accelerated motion way is

1. Calculate the initial position of the object.
2. Calculate the position of the object after 20 s of motion.

Solution 5

1. The initial position is obtained for t = 0. Thus,
   x(t) = 5 + 3t + 2t²
   x(0) = 5 + 3 ∙ 0 + 2 ∙ 0²
   = 5 + 0 + 0
   = 5
   This means the object was initially 5 m ahead of the origin.
2. The position after 20 s is calculated by taking t = 20. Thus,
   x(t) = 5 + 3t + 2t²
   x(20) = 5 + 3 ∙ 20 + 2 ∙ 20²
   = 5 + 3 ∙ 20 + 2 ∙ 400
   = 5 + 60 + 800
   = 865
   Hence, after 20 s of motion the object will be 865 m ahead of the origin.

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7. Continuing learning formulas - read our next math tutorial: Types of Formulas. Rearranging Formulas

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