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6.04 Substitution

Lesson

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:

x12345
y47101316

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+1Substitute x=1 into the equation
\displaystyle =\displaystyle 18+1Simplify the product
\displaystyle =\displaystyle 19Evaluate

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+1Write the equation
\displaystyle =\displaystyle 3 \times 20 + 1Substitute x=20 into the equation
\displaystyle =\displaystyle 60 + 1Simplify the product
\displaystyle =\displaystyle 61Evaluate

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

\displaystyle y\displaystyle =\displaystyle 3x + 1Write the equation
\displaystyle =\displaystyle 3 \times 105 + 1Substitute x=105 into the equation
\displaystyle =\displaystyle 315 + 1Simplify the product
\displaystyle =\displaystyle 316Evaluate

Examples

Example 1

Find the value of 9+m when m=3.

Worked Solution
Create a strategy

We can substitute m=3 into the equation by replacing m with 3.

Apply the idea
\displaystyle 9+m\displaystyle =\displaystyle 9+3Substitute m=3 into the equation
\displaystyle =\displaystyle 12Evaluate the addition

Example 2

Find the value of \dfrac{u}{9} when u=54.

Worked Solution
Create a strategy

We can substitute u=54 into the equation by replacing u with 54.

Apply the idea
\displaystyle \dfrac{u}{9}\displaystyle =\displaystyle \dfrac{54}{9}Substitute u=54 into the equation
\displaystyle =\displaystyle 6Evaluate the quotient

Example 3

Evaluate 6x+4y+6 when x=3 and y=5.

Worked Solution
Create a strategy

We can evaluate the expression by substituting in the values for x and y.

Apply the idea
\displaystyle 6x+4y+6\displaystyle =\displaystyle 6\times 3 +4\times 5+6Substitute the values of\,x and y
\displaystyle =\displaystyle 18 + 20 +6Perform multiplication
\displaystyle =\displaystyle 44Evaluate the addition
Idea summary

Substitution is the replacing of the pronumerals with numbers.

A key part of substitution is understanding the equation and identifying which pronumeral to substitute.

Outcomes

VCMNA252

Create algebraic expressions and evaluate them by substituting a given value for each variable

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