Putting numbers back in

For­mu­las are every­where: the perime­ter of a rec­tan­gle, the cost of sev­eral note­books, the dis­tance trav­elled in a given time. Each for­mula is writ­ten with let­ters so that it works for every case. The moment you have an actual rec­tan­gle or an actual jour­ney, you put its num­bers in place of the let­ters and work out the answer. That step is called sub­sti­tu­tion, and it is one of the most used skills in all of alge­bra.

What sub­sti­tu­tion is#

Sub­sti­tu­tion. The method of replac­ing the let­ters (lit­eral num­bers) in an expres­sion by numer­i­cal val­ues is called sub­sti­tu­tion.

An expres­sion like $5x$ is not an answer; it is a recipe for find­ing one. It says: take $x$ and mul­ti­ply it by $5$. You can­not fol­low the recipe while $x$ is still a let­ter. Give $x$ a value and the recipe runs, turn­ing the expres­sion into one plain num­ber.

Exam­ple 1#

Find the value of $2x$ when $x = -1$.

$$2x = 2 \times (-1) = -2$$

Write $(-1)$ in the place of $x$, brack­ets and all, and then mul­ti­ply.

One let­ter, one or two steps#

Exam­ple 2#

Find the value of $5x$ when $x = 3$.

$$5x = 5 \times 3 = 15$$

The mul­ti­pli­ca­tion sign is hid­ing in $5x$. Writ­ing $5 \times 3$ brings it back into view. Miss it and you might write $53$ by mis­take.

Exam­ple 3#

Find the value of $3a + 2$ when $a = 4$.

$$\begin{aligned} 3a + 2 &= 3 \times 4 + 2 \\ &= 12 + 2 \\ &= 14 \end{aligned}$$

Mul­ti­pli­ca­tion is always done before addi­tion. If you added first, $4 + 2 = 6$ and $3 \times 6 = 18$, which is wrong.

The trap: $x^2$ is not $2x$#

$x^2$ means $x \times x$. It does not mean $2 \times x$. The small raised $2$ counts how many $x$s are mul­ti­plied together; a num­ber in front tells you what to mul­ti­ply $x$ by.

Exam­ple 4#

Find the val­ues of $x^2$ and $2x$ when $x = 5$.

$$x^2 = 5 \times 5 = 25 \qquad 2x = 2 \times 5 = 10$$

Same let­ter, same value, and two very dif­fer­ent answers.

At $x = 2$ the two hap­pen to agree, because $2 \times 2$ and $2 + 2$ are both $4$. For every larger value, $x^2$ is big­ger, and the gap keeps grow­ing.

Exam­ple 5#

Find the value of $2x^2$ when $x = 3$.

$$\begin{aligned} 2x^2 &= 2 \times 3 \times 3 \\ &= 2 \times 9 \\ &= 18 \end{aligned}$$

Only the $x$ is squared, not the $2$. If the whole of $2x$ were meant to be squared, it would be writ­ten $(2x)^2$, and that is $6 \times 6 = 36$.

More than one let­ter#

An expres­sion can hold sev­eral let­ters, each with its own value. Put them all in, then sim­plify.

Exam­ple 6#

Find the value of $2a + 3b$ when $a = 1$ and $b = 2$.

$$\begin{aligned} 2a + 3b &= 2 \times 1 + 3 \times 2 \\ &= 2 + 6 \\ &= 8 \end{aligned}$$

Exam­ple 7: A for­mula#

The perime­ter of a rec­tan­gle is $P = 2(l + b)$. Find $P$ when $l = 7$ cm and $b = 4$ cm.

$$\begin{aligned} P &= 2(l + b) \\ &= 2 \times (7 + 4) \\ &= 2 \times 11 \\ &= 22 \end{aligned}$$

The perime­ter is 22 cm. The bracket was worked out first, then the mul­ti­ply­ing.

When the value is neg­a­tive#

A minus sign is very easy to lose. Brack­ets keep it where it belongs. Always put brack­ets round a neg­a­tive value when you sub­sti­tute it.

Exam­ple 8#

Find the value of $5 - x$ when $x = -2$.

$$\begin{aligned} 5 - x &= 5 - (-2) \\ &= 5 + 2 \\ &= 7 \end{aligned}$$

Tak­ing away means mov­ing down the num­ber line. Now $-2$ sits two steps below zero, and tak­ing away some­thing below zero pushes you back up. That is why $5 - (-2)$ is $7$ and not $3$.

Exam­ple 9#

Find the value of $a^2 - b$ when $a = -3$ and $b = 4$.

$$\begin{aligned} a^2 - b &= (-3) \times (-3) - 4 \\ &= 9 - 4 \\ &= 5 \end{aligned}$$

A neg­a­tive num­ber times a neg­a­tive num­ber is pos­i­tive, so $(-3)^2 = 9$. With­out the brack­ets, $-3^2$ would be read as $-(3 \times 3) = -9$, a dif­fer­ent num­ber.

Exam­ple 10#

Find the value of $x^2 - 3xy + 2y^2$ when $x = 2$ and $y = -1$.

$$\begin{aligned} x^2 - 3xy + 2y^2 &= (2)^2 - 3(2)(-1) + 2(-1)^2 \\ &= 4 + 6 + 2 \\ &= 12 \end{aligned}$$

Here $-3 \times 2 \times (-1) = 6$, and $2 \times (-1)^2 = 2 \times 1 = 2$.

The safe order#

1. Copy the expres­sion out.
2. Replace each let­ter with its value. Put brack­ets round any neg­a­tive value.
3. Put back the mul­ti­pli­ca­tion signs that were hid­den.
4. Sim­plify: brack­ets first, then pow­ers, then mul­ti­pli­ca­tion and divi­sion, then addi­tion and sub­trac­tion.
5. Check that no let­ter is left behind.

Prac­tice#

1. Find the value of $4x$ when $x = 2$.
2. Find the value of $3y$ when $y = -4$.
3. Find the value of $2a + 5$ when $a = 3$.
4. Find the value of $x^2 + 5$ when $x = 6$.
5. Find the value of $3m + 2n$ when $m = 2$ and $n = 5$.
6. Find the value of $10 - y$ when $y = -3$.
7. Find the value of $3(a + b)$ when $a = 2$ and $b = -5$.
8. Find the value of $p^2 - 2pq + q^2$ when $p = 3$ and $q = -2$.
9. The area of a tri­an­gle is $\displaystyle A = \frac{1}{2}bh$. Find $A$ when $b = 12$ cm and $h = 7$ cm.

Answers#