Lesson 2: Simple substitution

Introduction

Time to start building your substitution skills! You'll be dealing with simple expressions this lesson which are quite easy to evaluate. But it's not just about getting the answers right. The knowledge and skills you gain here will be used throughout the whole of algebra!

SECTION A

We're going to start with simple expressions that involve ONLY addition and subtraction.

Example 1

Evaluate \(~x+5~\),   if \(~x=3~\)

Substitute and evaluate: \(~3+5=8~\)

Always write out the expression exactly as it is in the question but with numbers in place of letters. Then work out the answer. In this case, write \(~3+5=8~\), not just \(~8~\).

Example 2

Evaluate \(~a-7~\),   if \(~a=20~\)

Substitute and evaluate: \(~20-7=13~\)

Example 3

Evaluate \(~12-b~\),   if \(~b=3~\)

Substitute and evaluate: \(~12-3=9~\)

In example 2, the letter came first whereas in example 3, the letter was at the end. When you replace a letter, it's really important that you put the corresponding number exactly where the letter was!

Example 4

Evaluate \(~p+q~\),   if \(~p=11~\) and \(~q=4~\)

Substitute and evaluate: \(~11+4=15~\)

There can be more than one letter in the expression but you have to be given the value of each letter, of course!

Example 5

Evaluate \(~3+a-b+c~\),   if \(~a=9~\), \(~b=5~\) and \(~c=7~\)

Substitute and evaluate: \(~3+9-5+7=14~\)

An expression can have any number of letters (and a number) separated by \(~+~\) or \(~-~\) signs. Work them out from left to right!

Complete these practice questions perfectly and you'll be ready to move on.

Practise to master

SECTION A

Evaluate the following expressions (set out your answers correctly).

01)  \(m+12~\)   (\(m=9\))

\(9+12=21\)

02)  \(3+a~\)   (\(a=22\))

\(3+22=25\)

03)  \(t-1~\)   (\(t=4\))

\(4-1=3\)

04)  \(13-w~\)   (\(w=6\))

\(13-6=7\)

05)  \(d+e~\)   (\(d=17\) ,  \(e=12\))

\(17+12=29\)

06)  \(b+6~\)   (\(b=7\))

\(7+6=13\)

07)  \(9+f~\)   (\(f=8\))

\(9+8=17\)

08)  \(z-2~\)   (\(z=5\))

\(5-2=3\)

09)  \(8-y~\)   (\(y=3\))

\(8-3=5\)

10)  \(g+h~\)   (\(g=0\) ,  \(h=89\))

\(0+89=89\)

11)  \(e+11~\)   (\(e=30\))

\(30+11=41\)

12)  \(19+u~\)   (\(u=4\))

\(19+4=23\)

13)  \(w-1~\)   (\(w=8\))

\(8-1=7\)

14)  \(17-b~\)   (\(b=8\))

\(17-8=9\)

15)  \(j-k~\)   (\(j=53\) ,  \(k=22\))

\(53-22=31\)

16)  \(g+h-4~\)   (\(g=20\) ,  \(h=1\))

\(20+1-4=17\)
Working from left to right, \(~20+1=21~\) and \(~21-4=17~\)

17)  \(6-m+n~\)   (\(m=5\) ,  \(n=24\))

\(6-5+24=25\)
Working from left to right, \(~6-5=1~\) and \(~1+24=25~\)

18)  \(x-y+z-2~\)   (\(x=34\) ,  \(y=19\) ,  \(z=5\))

\(34-19+5-2=18\)
Working from left to right, \(~34-19=15~\), \(~15+5=20~\) and \(~20-2=18~\)

19)  \(45-a+b+c~\)   (\(a=16\) ,  \(b=1\) ,  \(c=7\))

\(45-16+1+7=37\)
Working from left to right, \(~45-16=29~\), \(~29+1=30~\) and \(~30+7=37~\)

20)  \(u-v+x-y~\)   (\(u=60\) ,  \(v=35\) ,  \(x=9\) ,  \(y=12\))

\(60-35+9-12=22\)
Working from left to right, \(~60-35=25~\), \(~25+9=34~\) and \(~34-12=22~\)

SECTION B

We're now going to look at simple expressions that involve ONLY multiplication and division.

How to write multiplication in algebra

We don't use a \(~\times~\) sign between letters, or between a number and a letter! We just write the letters next to each other (or the number and the letter next to each other).

So  \(~3\times a~\) is written \(~3a~\)

You'll never see \(~a3~\) because the number always goes in front of any letter(s).

And  \(~5\times x\times y\times z~\) is written \(~5xyz~\),  etc.

Letters should be written in alphabetical order.

How to write division in algebra

Division is not written with a \(~\div~\) sign in algebra, it is written as a fraction.

So \(~x\div 5~\) is written \(~\Large\frac{x}{5}~\)

Example 1

Evaluate \(~4x~\),   if \(~x=6~\)

Substitute and evaluate: \(~4\times 6=24~\)

Note that as soon as the \(~x~\) was replaced with a number, the \(~\times~\) sign had to go back in. Otherwise, we'd have \(~46~\) which is confusing and wrong!

Example 2

Evaluate \(~ab~\),   if \(~a=5~\) and \(~b=9~\)

Substitute and evaluate: \(~5\times 9=45~\)

Example 3

Evaluate \(~\Large\frac{n}{3}~\),   if \(~n=15~\)

Substitute and evaluate: \(~\Large\frac{15}{3}~\normalsize=~5\)

Example 4

Evaluate \(~\Large\frac{32}{k}~\),   if \(~k=8~\)

Substitute and evaluate: \(~\Large\frac{32}{8}~\normalsize=~4\)

Example 5

Evaluate \(~abc~\),   if \(~a=2~\), \(~b=3~\) and \(~c=5~\)

Substitute and evaluate: \(~2\times 3\times 5~=~30\)

Example 6

Evaluate \(~\Large\frac{3x}{2y}~\),   if \(~x=4~\) and \(~y=3~\)

Substitute and evaluate: \(~\Large\frac{3\times 4}{2\times 3}~\normalsize=\Large\frac{12}{6}~\normalsize=~2\)

This is important! If there's stuff to work out in the numerator of a fraction, work it out and write your answer as a numerator. If there's stuff to work out in the denominator of a fraction, work it out and write your answer as a denominator.

Do the actual division last!

Again, do a perfect job of these questions to ensure that you are truely ready to move on.

Practise to master

SECTION B

Evaluate the following expressions (set out your answers correctly).

01)  \(10e~\)   (\(e=8\))

\(10\times 8=80\)

02)  \(ab~\)   (\(a=5\) ,  \(b=7\))

\(5\times 7=35\)

03)  \(\Large\frac{k}{4}~\)   (\(k=44\))

\(\Large\frac{44}{4}\normalsize=11\)

04)  \(\Large\frac{36}{s}~\)   (\(s=12\))

\(\Large\frac{36}{12}\normalsize=3\)

05)  \(\Large\frac{u}{v}~\)   (\(u=64\) ,  \(v=4\))

\(\Large\frac{64}{4}\normalsize=16\)

06)  \(6y~\)   (\(y=9\))

\(6\times 9=54\)

07)  \(pq~\)   (\(p=1\) ,  \(q=37\))

\(1\times 37=37\)

08)  \(\Large\frac{c}{8}~\)   (\(c=72\))

\(\Large\frac{72}{8}\normalsize=9\)

09)  \(\Large\frac{4}{v}~\)   (\(v=4\))

\(\Large\frac{4}{4}\normalsize=1\)

10)  \(\Large\frac{s}{t}~\)   (\(s=93\) ,  \(t=3\))

\(\Large\frac{93}{3}\normalsize=31\)

11)  \(7jk~\)   (\(j=2\),   \(k=10\))

\(7\times 2\times 10=140\)

12)  \(xyz~\)   (\(x=3\) ,  \(y=5\) ,  \(z=2\))

\(3\times 5\times 2=30\)

13)  \(\Large\frac{2u}{5}~\)   (\(u=30\))

\(\Large\frac{2\times 30}{5}~\normalsize=\Large\frac{60}{5}~\normalsize=~12\)
I normally write each line of workings underneath the last but with fractions I tend to work sideways.

14)  \(\Large\frac{45}{pq}~\)   (\(p=3\) ,  \(q=5\))

\(\Large\frac{45}{3\times 5}~\normalsize=\Large\frac{45}{15}~\normalsize=~3\)

15)  \(\Large\frac{3x}{2n}~\)   (\(x=12\) ,  \(n=6\))

\(\Large\frac{3\times 12}{2\times 6}~\normalsize=\Large\frac{36}{12}~\normalsize=~3\)

16)  \(11fgh~\)   (\(f=5\),   \(g=2\),   \(h=2\))

\(11\times 5\times 2\times 2=220\)

17)  \(mnpq~\)   (\(m=6\) ,  \(n=8\) ,  \(p=1\) ,  \(q=2\))

\(6\times 8\times 1\times 2=96\)

18)  \(\Large\frac{2z}{3xy}~\)   (\(x=2\) ,  \(y=2\),   \(z=24\))

\(\Large\frac{2\times 24}{3\times 2\times 2}~\normalsize=\Large\frac{48}{12}~\normalsize=~4\)

19)  \(\Large\frac{ab}{3c}~\)   (\(a=6\) ,  \(b=4\) ,  \(c=2\))

\(\Large\frac{6\times 4}{3\times 2}~\normalsize=\Large\frac{24}{6}~\normalsize=~4\)

20)  \(\Large\frac{2uv}{st}~\)   (\(u=5\) ,  \(v=8\) ,  \(s=4\) ,  \(t=4\))

\(\Large\frac{2\times 5\times 8}{4\times 4}~\normalsize=\Large\frac{80}{16}~\normalsize=~5\)

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