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8.04 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 4th value in the pattern, and so on. The y-values represent the numbers in the pattern, so the 1st value is 4 and the 4th value is 13.

We are trying to find the next value, which in this case is the 6th 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=6 into the equation
\displaystyle =\displaystyle 18+1Simplify the product
\displaystyle =\displaystyle 19Evaluate

We can see that the 6th number in the pattern is 19.

Now we could have found this value by adding 3 to the 5th number, since the numbers in the pattern go up by 3 each step. But what if we are asked to find the 20th (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 20th number in the pattern (x=20):

\displaystyle y\displaystyle =\displaystyle 3x-5Write 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-5Write the equation
\displaystyle =\displaystyle 3\times 105 -5Substitute 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 the multiplications
\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

MA4-9NA

operates with positive-integer and zero indices of numerical bases

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