Ideas
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:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 4 | 7 | 10 | 13 | 16 |
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:
| \displaystyle y | \displaystyle = | \displaystyle 3x+1 | |
| \displaystyle = | \displaystyle 3\times 6+1 | Substitute x=1 into the equation | |
| \displaystyle = | \displaystyle 18+1 | Simplify the product | |
| \displaystyle = | \displaystyle 19 | Evaluate |
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):
| \displaystyle y | \displaystyle = | \displaystyle 3x-5 | Write the equation |
| \displaystyle = | \displaystyle 3\times 20+1 | Substitute x=20 into the equation | |
| \displaystyle = | \displaystyle 60+1 | Simplify the product | |
| \displaystyle = | \displaystyle 61 | Evaluate |
.. and the 105\text{th} number (x=105):
| \displaystyle y | \displaystyle = | \displaystyle 3x-5 | Write the equation |
| \displaystyle = | \displaystyle 3\times 105 -5 | Substitute x=105 into the equation | |
| \displaystyle = | \displaystyle 315+1 | Simplify the product | |
| \displaystyle = | \displaystyle 316 | Evaluate |
Examples
Example 1
Find the value of 9+m when m=3.
Example 2
Find the value of \dfrac{u}{9} when u=54.
Example 3
Evaluate 6x+4y+6 when x=3 and y=5.
Substitution is the replacing of the pronumerals with numbers.
A key part of substitution is understanding the equation and identifying which pronumeral to substitute.