6.04 Substitution

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

Exploration

Consider the following table of values:

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5
$y$y	$4$4	$7$7	$10$10	$13$13	$16$16

We can construct an equation describing the relationship between $x$x and $y$y:

$y=3x+1$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 $x$x-values represent the position in the pattern. For example, the $x=1$x=1 column represents the $1$1st position, the $x=4$x=4 column represents the $4$4th value in the pattern, and so on.

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

We are trying to find the next value, which in this case is the $6$6th value. In other words, we want to find:

"What is the value of $y$y when the value of $x$x is $6$6?"

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

$y$y	$=$=	$3x+1$3x+1
$y$y	$=$=	$3\times6+1$3×6+1	(substitute $x=6$x=6 into the equation)
$y$y	$=$=	$18+1$18+1	(simplify the product)
$y$y	$=$=	$19$19	(evaluate the addition)

 

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

Now we could have found this value by adding $3$3 to the $5$5th number, since the numbers in the pattern go up by $3$3 each step. But what if we are asked to find the $20$20th (or the $105$105th) number in the pattern? We don't want to add $3$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$20th number in the pattern ($x=20$x=20):

$y$y	$=$=	$3x-5$3x−5
$y$y	$=$=	$3\times20+1$3×20+1
$y$y	$=$=	$61$61

... and the $105$105th number ($x=105$x=105):

$y$y	$=$=	$3x-5$3x−5
$y$y	$=$=	$3\times105+1$3×105+1
$y$y	$=$=	$316$316

Worked examples

EXAMPLE 1

Consider the following equation.

$P=6Q+2$P=6Q+2

What is the value of $P$P if $Q=4$Q=4?

Think: We are trying to solve the equation for $P$P and we are given the value of $Q$Q so we can substitute in $4$4 for $Q$Q to find the value of $P$P.

Do: Wherever $Q$Q appears in the equation we replace it with it's value of $4$4:

$P$P	$=$=	$6Q+2$6Q+2
$P$P	$=$=	$6\times4+2$6×4+2
$P$P	$=$=	$26$26

Reflect: The process of substitution is putting a number where the pronumeral is, we can do this because the pronumeral is equivalent to the number.

We had two unknown pronumerals and we were given the value of one, which we used to find the value of the other. Notice that this value of $P$P is only when $Q=4$Q=4. As $Q$Q changes so does $P$P, so we can also think of this problem in terms of a table of values.

EXAMPLE 2

Inside a room there are $4$4 dogs and $3$3 birds. The dogs have $4$4 legs each, and the birds have $2$2 legs each. How many legs in total are there in this room?

Think: We create an expression relating the number of dogs, the number of birds, and the number of legs. Then we can substitute values into our expression to find the total.

Do: Let the total number of legs be $T$T, the number of dogs be $D$D, and the number of birds be $B$B.

Every dog will contribute $4$4 legs to the total and every bird will add $2$2, so our equation will be:

$T=4D+2B$T=4D+2B

Let's substitute $D=4$D=4 and $B=3$B=3 into the the equation above. In other words, replace $D$D with $4$4 and $B$B with $3$3:

$T$T	$=$=	$4D+2B$4D+2B
$T$T	$=$=	$4\times4+2\times3$4×4+2×3
$T$T	$=$=	$22$22

Reflect: We are now working with two pronumerals. An algebraic expression can contain any number of pronumerals and we can substitute any number of these values in at once.

Practice questions

Question 1

Question 2

Question 3