6.04 Substitution

After building an algebraic expression we can solve it by substitution, where we replace pronumerals with numeric values.

Consider the following table of values:

We can construct an equation describing the relationship between x and y:y=3x+1

What is the next number in the pattern?

We can solve this problem by using substitution. A key part of substitution is understanding the equation and identifying which pronumeral to substitute.

In this table of values, the x-values represent the position in the pattern. For example, the x=1 column represents the 1st position, the x=4 column represents the 4\text{th} value in the pattern, and so on. The y-values represent the numbers in the pattern, so the 1\text{st} value is 4 and the 4\text{th} value is 13.

We are trying to find the next value, which in this case is the 6\text{th} value. In other words, we want to find what the value of y is when the value of x is 6.

In an algebraic expression, the term 3x means 3\times x. So if we substituted x=4 into the equation then the term is equal to 3\times 4= 12 and not 34.

Let's now perform the substitution, using x=6:

We can see that the 6\text{th} number in the pattern is 19.

Now we could have found this value by adding 3 to the 5\text{th} number, since the numbers in the pattern go up by 3 each step. But what if we are asked to find the 20\text{th} (or the 105\text{th}) number in the pattern? We don't want to add 3 twenty (or one hundred and five) times.

Substitution allows us to find the answer directly, no matter what number we choose. We can find the 20\text{th} number in the pattern (x=20):

.. and the 105\text{th} number (x=105):

Examples