Lesson 4: Substitution with negative numbers

Introduction

As the last two lessons progressed, the expressions you had to deal with became more and more complex, while the numbers you substituted into those expressions were always nice positive integers - you're welcome!

This lesson, you're going to be substituting negative values into expressions. I'm assuming you can add, subtract, multiply and divide with negative numbers. If you can't, you will struggle with every single algebra topic you come across. Seriously! Even if you're an expert with negative numbers, this lesson will introduce slight differences in the way your workings are written out.

I should point out that decimals and fractions can also be substituted into expressions. You will come across these in the lesson Application of formulae.

SECTION A

Addition and subtraction with negative numbers

When adding and subtracting with negative numbers, imagine a number line. For example, to evaluate \(~-5+9-2~\),  imagine standing on a number line at \(~-5~\),  move \(~9~\) places to the right (because it's plus \(9\)) then \(~2~\) places to the left (because it's minus \(2\)).  You land on \(+2~\) so the answer is \(~2~\).

You'll often have two signs right next to each other. For example, \(~3--8~\) means we're subtracting a minus number. In this case, change the two signs to a single sign using this rule: if the signs are the same, replace them with a single \(+~\); if they're different, replace them with a single \(-~\).  So \(~3--8=3+8=11~\).

Applying this to substitution

Example 1

Evaluate \(~5+x~\),   if \(~x=-7\)

Substitute: \(~5+-7\)

Replace double sign: \(~5-7\)

Imagine number line: \(~-2\)

Substituting means taking the \(~x~\) out and replacing it with \(~-7~\). The two signs right next to each other are different so we replace them with a single \(~-~\) sign.

Example 2

Evaluate \(~a-2~\),   if \(~a=-3\)

Substitute: \(~-3-2\)

Imagine number line: \(~-5\)

Again the \(~a~\) was taken out and replaced with \(~-3~\).

Example 3

Evaluate \(~p+q-1~\),   if \(~p=-7\)  and  \(~q=-4\)

Substitute: \(~-7+-4-1\)

Replace double sign: \(~-7-4-1\)

Imagine number line: \(~-12\)

Be extra careful with that first step. In fact, be extra careful with every step when negative numbers are involved!

Let's have a few practice questions.

Practise to master

SECTION A

Evaluate the following expressions using your negative number skills.

01)  \(x+16~\)   (\(x=-5\))

\(-5+16=11\)

02)  \(9+c~\)   (\(c=-20\))

\(9+-20\)
\(9-20\)
\(-11\)

03)  \(y-1~\)   (\(y=-7\))

\(-7-1=-8\)

04)  \(2-x~\)   (\(x=-3\))

\(2--3\)
\(2+3\)
\(5\)

05)  \(p+q~\)   (\(p=-12\) ,  \(q=-10\))

\(-12+-10\)
\(-12-10\)
\(-22\)

06)  \(x-y+3~\)   (\(x=-4\) ,  \(y=-9\))

\(-4--9+3\)
\(-4+9+3\)
\(8\)

07)  \(a+5-b~\)   (\(a=-15\) ,  \(b=-20\))

\(-15+5--20\)
\(-15+5+20\)
\(10\)

08)  \(1-u-v~\)   (\(u=13\) ,  \(v=-6\))

\(1-13--6\)
\(1-13+6\)
\(-6\)

09)  \(x+y+z~\)   (\(x=-7\) ,  \(y=-8\) ,  \(z=14\))

\(-7+-8+14\)
\(-7-8+14\)
\(-1\)

10)  \(a-b+c~\)   (\(a=-3\) ,  \(b=1\) ,  \(c=-21\))

\(-3-1+-21\)
\(-3-1-21\)
\(-25\)

SECTION B

Multiplication and division with negative numbers

When multiplying and dividing, follow the rule: if the signs are the same, the answer is positive and if the signs are different, the answer is negative. For example, \(~-5\times 3=-15~\) because \(~5\times 3=15~\) and the signs on the \(~5~\) and on the \(~3~\) are different so the answer must be negative.

Applying this to substitution

Example 1

Evaluate \(~3a~\),   if \(~a=-4\)

Substitute: \(~3(-4)~\)

Multiplication: \(~-12\)

So, here it is! When substituting a negative number that is to be multiplied, put it in brackets instead of using a \(~\times~\) sign. Remember, if something is placed directly in front of a bracket it means it multiplies what's in that bracket. BTW you can do this when substituting positive numbers too, if you like. I often do!

Example 2

Evaluate \(~xy~\),   if \(~x=-6\)  and  \(~y=-5\)

Substitute: \(~(-6)(-5)~\)

Multiplication: \(~30\)

Here I've substituted two negative numbers and put each of them in brackets. The signs are the same so the answer must be positive.

Example 3

Evaluate \(~\Large\frac{p}{q}~\),   if \(~p=18\)  and  \(~q=-6\)

Substitute: \(~\Large\frac{18}{-6}~\)

Division: \(~-3\)

\(18\div 6=3\) and the signs are different so the answer is negative.

Let's do some multiplication and division. Remember to put numbers you're substituting in brackets if they're multiplying or being multiplied by something.

Practise to master

SECTION B

Evaluate the following expressions.

01)  \(6d\)   (\(d=-4\))

\(6(-4)=-24\)
Because \(~6\times 4=24~\) and different signs ⇒ negative answer

02)  \(mn\)   (\(m=-5\) ,  \(n=-6\))

\((-5)(-6)=30\)
Because \(~5\times 6=30~\) and same signs ⇒ positive answer

03)  \(\Large\frac{t}{4}\)   (\(t=-28\))

\(\Large\frac{-28}{4}\normalsize~=~-7\)
Because \(~28\div 4=7~\) and different signs ⇒ negative answer

04)  \(\Large\frac{-48}{w}\)   (\(w=-16\))

\(\Large\frac{-48}{-16}\normalsize~=~3\)
Because \(~48\div 16=3~\) and same signs ⇒ positive answer

05)  \(\Large\frac{a}{b}\)   (\(a=-8\) ,  \(b=2\))

\(\Large\frac{-8}{2}\normalsize~=~-4\)
Because \(~8\div 2=4~\) and different signs ⇒ negative answer

06)  \(4pq\)   (\(p=-3\) ,  \(q=-5\))

\(4(-3)(-5)=4(15)=60\)
If you need to multiply three things together, multiply two of them together then multiply the answer by the third. It doesn't matter which two you pick to do first.

07)  \(xyz\)   (\(x=-2\) ,  \(y=-3\) ,  \(z=-4\))

\((-2)(-3)(-4)=(6)(-4)=-24\)
Again, multiply two at a time.

08)  \(\Large\frac{2u}{v}\)   (\(u=-15\) ,  \(v=-6\))

\(\Large\frac{2(-15)}{-6}\normalsize~=~\Large\frac{-30}{-6}\normalsize~=~5\)

09)  \(\Large\frac{30}{cd}\)   (\(c=-5\) ,  \(d=-2\))

\(\Large\frac{30}{(-5)(-2)}\normalsize~=~\Large\frac{30}{10}\normalsize~=~3\)

10)  \(\Large\frac{3x}{-2y}\)   (\(x=-8\) ,  \(y=-6\))

\(\Large\frac{3(-8)}{-2(-6)}\normalsize~=~\Large\frac{-24}{12}\normalsize~=~-2\)

SECTION C

Mixed operations with negative numbers

We'll be calling on BIDMAS again now as we're mixing addition and subtraction with multiplication and division.

Example 1

Evaluate \(~5-2x~\),   if \(~x=-8\)

Substitute: \(~5-2(-8)~\)

Multiplication: \(~5--16\)

Replace double signs: \(~5+16\)

Addition: \(~21\)

There are two ways of evaluating this:
\(~5~~~-2(-8)~=~5~~~~~~~~+16~=~21\)
\(~5-~~~2(-8)~=~5-~~~-16~=~5~+16~=~21\)
I use the latter in these tutorials as it's clearer, I think. This is not the place for a detailed discussion on this but it's worth mentioning.

Example 2

Evaluate \(~2a+\Large\frac{20}{b}~\),   if \(~a=-9\)  and  \(~b=-5\)

Substitute: \(~2(-9)+\Large\frac{20}{-5}~\)

Multiplication and division: \(~-18+-4~\)

Replace double sign: \(~-18-4\)

Addition and subtraction: \(~-22\)

Take your time and get these right. Not far to go now!

Practise to master

SECTION C

Evaluate the following expressions.

01)  \(1+8x\)   (\(x=-9\))

\(1+8(-9)\)
\(1+-72\)
\(1-72\)
\(-71\)

02)  \(5-ab\)   (\(a=-1\) ,  \(b=-4\))

\(5-(-1)(-4)\)
\(5-4\)
\(1\)

03)  \(2x+7y\)   (\(x=-3\) ,  \(y=-2\))

\(2(-3)+7(-2)\)
\(-6+-14\)
\(-6-14\)
\(-20\)

04)  \(6c-5de\)   (\(c=-4\) ,  \(d=3\) ,  \(e=-2\))

\(6(-4)-5(3)(-2)\)
\(-24-5(-6)\)
\(-24--30\)
\(-24+30\)
\(6\)

05)  \(2x+5y-3z\)   (\(x=-10\) ,  \(y=-3\) ,  \(z=2\))

\(2(-10)+5(-3)-3(2)\)
\(-20+-15-6\)
\(-20-15-6\)
\(-41\)

06)  \(2+\Large\frac{t}{6}\)   (\(t=-42\))

\(2+\Large\frac{-42}{6}\)
\(2+-7\)
\(2-7\)
\(-5\)

07)  \(5x-\Large\frac{52}{y}\)   (\(x=2\) ,  \(y=-13\))

\(5(2)-\Large\frac{52}{-13}\)
\(10--4\)
\(10+4\)
\(14\)

08)  \(\Large\frac{a}{4}\normalsize-9b\)   (\(a=-12\) ,  \(b=-3\))

\(\Large\frac{-12}{4}\normalsize-9(-3)\)
\(-3--27\)
\(-3+27\)
\(24\)

09)  \(\Large\frac{40}{x}\normalsize+\Large\frac{y}{3}\)   (\(x=-5\) ,  \(y=-15\))

\(\Large\frac{40}{-5}\normalsize+\Large\frac{-15}{3}\)
\(-8+-5\)
\(-8-5\)
\(-13\)

10)  \(\Large\frac{r}{8}\normalsize+6s-\Large\frac{t}{2}\)   (\(r=-32\) ,  \(s=-2\) ,  \(t=-20\))

\(\Large\frac{-32}{8}\normalsize+6(-2)-\Large\frac{-20}{2}\)
\(-4+-12--10\)
\(-4-12+10\)
\(-6\)

SECTION D

Expressions in brackets with negative numbers

Terms within expressions are separated by \(~+~\) and/or \(~-~\) signs.
Expressions within terms are generally put in brackets.

Example 1

Evaluate \(~3(1+2x)-5y~\),   if \(~x=-4~\) and \(~y=-2~\)

Substitute: \(~3(1+2(-4))-5(-2)\)

Expressions in brackets: \(~3(1+-8)-5(-2)\)

Double signs: \(~3(1-8)-5(-2)\)

Expressions in brackets: \(~3(-7)-5(-2)\)

Multiplication: \(~-21--10\)

Double signs: \(~-21+10\)

Finish off: \(~-11\)

If something is right next to a bracket, it multiplies what's in that bracket so there's no need for \(~\times~\) signs. We really don't like \(~\times~\) signs, do we!

Example 2

Evaluate \(~2(x-1)(2+y)~\),   if \(~x=-2~\) and \(~y=-5~\)

Substitute: \(~2(-2-1)(2+-5)~\)

Double signs: \(~2(-2-1)(2-5)~\)

Expressions in brackets: \(~2(-3)(-3)~\)

Multiplication: \(~2(9)\)

Multiplication: \(~18\)

Two brackets right next to each other with nothing in between means they're multiplied together. Again, no need to use \(~\times~\) signs.

Example 3

Evaluate \(~\Large\frac{3(a-1)}{7+b}~\),   if \(~a=-3\)  and \(~b=-5\)

Substitute: \(~\Large\frac{3(-3-1)}{7+-5}~\)

Double signs: \(~\Large\frac{3(-3-1)}{7-5}~\)

Expressions in brackets: \(~\Large\frac{3(-4)}{7-5}~\)

Top and bottom of fraction: \(~\Large\frac{-12}{2}~\)

Division: \(~-6\)

Nothing too difficult here. You could almost say the objective of this exercise is to get you used to using brackets in place of \(~\times~\) signs. It does make evaluating complicated expressions a bit easier.

Practise to master

SECTION D

Evaluate the following expressions.

01)  \(2(a-6)\)   (\(a=-4\))

\(2(-4-6)\)
\(2(-10)\)
\(-20\)

02)  \(3(x+2)-9\)   (\(x=-5\))

\(3(-5+2)-9\)
\(3(-3)-9\)
\(-9-9\)
\(-18\)

03)  \(4(2x+5)+7\)   (\(x=-3\))

\(4(2(-3)+5)+7\)
\(4(-6+5)+7\)
\(4(-1)+7\)
\(-4+7\)
\(3\)

04)  \(2\Big(\Large\frac{x}{2}\normalsize-7\Big)+3y\)   (\(x=-14\) ,  \(y=-1\))

\(2\Big(\Large\frac{-14}{2}\normalsize-7\Big)+3(-1)\)
\(2(-7-7)+3(-1)\)
\(2(-14)+3(-1)\)
\(-28+-3\)
\(-28-3\)
\(-31\)

05)  \(3(a+3)(b-5)\)   (\(a=-12\) ,  \(b=-2\))

\(3(-12+3)(-2-5)\)
\(3(-9)(-7)\)
\(3(63)\)
\(189\)

06)  \(\Large\frac{x+4}{y-3}\)   (\(x=-9\) ,  \(y=-2\))

\(\Large\frac{-9+4}{-2-3}\normalsize~=~\Large\frac{-5}{-5}\normalsize~=~1\)

07)  \(\Large\frac{3a-1}{b+12}\)   (\(a=-5\) ,  \(b=-4\))

\(\Large\frac{3(-5)-1}{-4+12}\normalsize~=~\Large\frac{-15-1}{-4+12}\normalsize~=~\Large\frac{-16}{8}\normalsize~=~-2\)

08)  \(\Large\frac{6(x+2)}{y-2}\)   (\(x=-8\) ,  \(y=-1\))

\(\Large\frac{6(-8+2)}{-1-2}\normalsize~=~\Large\frac{6(-6)}{-1-2}\normalsize~=~\Large\frac{-36}{-3}\normalsize~=~12\)

09)  \(\Large\frac{2(3x+5)}{y+2}\)   (\(x=-2\) ,  \(y=-4\))

\(\Large\frac{2(3(-2)+5)}{-4+2}\normalsize~=~\Large\frac{2(-6+5)}{-4+2}\normalsize~=~\Large\frac{2(-1)}{-4+2}\normalsize~=~\Large\frac{-2}{-2}\normalsize~=~1\)

10)  \(\Large\frac{2(1-3a)}{b-3}\)   (\(a=-5\) ,  \(b=-1\))

\(\Large\frac{2(1-3(-5))}{-1-3}\normalsize~=~\Large\frac{2(1--15)}{-1-3}\normalsize~=~\Large\frac{2(1+15)}{-1-3}\normalsize~=~\Large\frac{2(16)}{-1-3}\normalsize~=~\Large\frac{32}{-4}\normalsize~=~-8\)

SECTION E

Indices with negative numbers

If you've got to this point, you must be good! I mean, you must be really good!

Example 1

Evaluate \(~x^2~\),   if \(~x=-4~\)

Substitute: \(~(-4)^2\)

This means the square of minus 4: \(~(-4)(-4)=16\)

With indices, as with multiplication, put the substituted value in brackets. A negative number raised to an even power will always give a positive answer.

Example 2

Evaluate \(~-x^2~\),   if \(~x=-4~\)

Substitute: \(~-(-4)^2\)

This means minus the square of minus 4: \(~-(-4)(-4)=-16\)

It might make it easier to rewrite the question as \(~-1x^2~\) so it's clear there's an index and a multiplication.

Example 3

Evaluate \(~x^3~\),   if \(~x=-4~\)

Substitute: \(~(-4)^3\)

This means the cube of minus 4: \(~(-4)(-4)(-4)=(-4)(16)=-64\)

You could, of course, work out \(~4^3~\) and then put a minus sign in front of the answer. A negative number raised to an odd power will always give a negative answer.

Well, I think that pretty much covers it!

Practise to master

SECTION E

Evaluate the following expressions.

01)  \(x^2-4\)   (\(x=-3\))

\((-3)^2-4\)
\(9-4\)
\(5\)

02)  \(-x^2+7\)   (\(x=-5\))

\(-(-5)^2+7\)
\(-25+7\)
\(-18\)

03)  \(2x^2-3x+2\)   (\(x=-2\))

\(2(-2)^2-3(-2)+2\)
\(2(4)-3(-2)+2\)
\(8--6+2\)
\(8+6+2\)
\(16\)
You'll use this skill for plotting quadratic graphs.

04)  \(-x^2+5x-3\)   (\(x=-1\))

\(-(-1)^2+5(-1)-3\)
\(-(1)+5(-1)-3\)
\(-1+-5-3\)
\(-1-5-3\)
\(-9\)

05)  \(\Large\frac{-b-\sqrt{b^2-4ac}}{2a}\)   (\(a=5\) ,  \(b=-8\) ,  \(c=-4\))

\(\large\frac{-(-8)-\sqrt{(-8)^2-4(5)(-4)}}{2(5)}\)
\(\large\frac{-(-8)-\sqrt{64-4(5)(-4)}}{2(5)}\)
\(\large\frac{-(-8)-\sqrt{64--80}}{2(5)}\)
\(\large\frac{-(-8)-\sqrt{64+80}}{2(5)}\)
\(\large\frac{8-\sqrt{144}}{10}\)
\(\large\frac{8-12}{10}\normalsize~=~\large\frac{-4}{10}\normalsize~=~-0.4\)
This formula (with the one below) is used to solve tricky quadratic equations.

06)  \(\Large\frac{-b+\sqrt{b^2-4ac}}{2a}\)   (\(a=6\) ,  \(b=-19\) ,  \(c=3\))

\(\large\frac{-(-19)+\sqrt{(-19)^2-4(6)(3)}}{2(6)}\)
\(\large\frac{-(-19)+\sqrt{361-4(6)(3)}}{2(6)}\)
\(\large\frac{19+\sqrt{361-72}}{12}\)
\(\large\frac{19+\sqrt{289}}{12}\)
\(\large\frac{19+17}{12}\normalsize~=~\large\frac{36}{12}\normalsize~=~3\)

07)  \(x^{56}\)   (\(x=-1\))

\((-1)^{56}~=~1\)
A minus number raised to an even power gives a positive answer.

08)  \(-x^{79}\)   (\(x=-1\))

\(-(-1)^{79}~=~-(-1)~=~1\)
A minus number raised to an odd power gives a negative answer. Then we had another minus which made the final answer positive.

09)  \(x^3-x^2+4x-3\)   (\(x=-1\))

\((-1)^3-(-1)^2+4(-1)-3\)
\(-1-1+-4-3\)
\(-1-1-4-3\)
\(-9\)
You'll use this skill for plotting cubic graphs.

10)  \(-x^3+3x^2-1\)   (\(x=-2\))

\(-(-2)^3+3(-2)^2-1\)
\(--8+3(4)-1\)
\(8+12-1\)
\(19\)

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