Essentials of Algebra

Evaluation 1

Evaluate the following expressions when s = 6 and t = −1.

(a) 2(s + t)

(b) s + t2

(c) st + 3t

(d) t(s + 3)

Terms 1

Identify the terms in the expression

34 3 27k h k

.

Terms 3

In the expression 24 3 2a a b

(a) The coefficient of b is 2.

(b) a2 has no coefficient.

(c) The coefficient of a2 is 1.

(d) The coefficient of a2 is −1.

(e) 3 is a constant term.

(f) 3 − 2b is a term.

(g) a2 is a term.

(h) 4a and − a2 are like terms.

True or false?

Simplifying 1

Simplifying 2

Simplify the following expressions by collecting like terms.

(a) k + 3k

(b) 2m + m2 − m

(c) 6xy − 3x − 2x + y

(d) 2ab − a2b − 5ba

(e) 4a + 2A + 3 − 1

(f) 4pq − 3q + pq +3q

Simplifying 4

Which of the following expressions are equivalent?

(a) xy + 3x − y

(b) xy − y + 3x

(c) y + xy + 3x

(d) y − xy + 3x

(e) 3x + yx − y

Simplifying 5

Write the following terms in their simplest forms.

(a) a × 2 × 3 × b × a

(b) 2a × 4ab

(c) t10 × 3t5

(d) a × (−b) × (−b)

(e) (−2) × g × 3h × (−g) × 7

Simplifying 7

Identify the terms in the following expressions and then simplify.

(a) 5a × 2b − (−a) × 4 × b

(b) 2 × (−7P) × P + Q × 8 × P × (−2)

Multiplying out brackets 1

Multiplying out brackets 2

Multiply out the brackets in the following expressions.

(a) 2a(3a − 6)

(b) 2(x + x2)y

(c) −2(4 − 2a)

(d) a(b + c) + b(a − c)

(e) 2a − (b − a)

More simplification

True or false?

(a) k − 3k = 2k

(b) a × b2 × (−2a) = −2a2b2

(c) 4 − (2 − x) = 2 + x

(d) a(b + c) − b(a − c) − c(b + a) = 0

2 6

2

A

Algebraic fractions 1

Expand the following algebraic fractions and then simplify where possible.

(a)

(b)25 3p p q pq

p

Think of a number 1

Start with a number

Start with n

Think of a number

Double it

Add 1

Multiply by 5

Subtract 3

Subtract ten times the number you first thought of

And the answer is…

Think of a number 3

Start with a number Start with n

Think of a number

Double it

Subtract 3

Multiply by 2

Add 10

Divide by 4

Subtract 1

And the answer is…

Proofs 1

Prove that the product of an even number and a multiple of 3 is always divisible by 6. (Hint: Write the even number as 2a and the multiple of 3 as 3b, where a and b are integers.)

Proofs 3

Do you think that the sum of an even number and a multiple of 3 is divisible by 6?

Find a counterexample, that is, find two numbers, an even number and a multiple of 3, whose sum is not divisible by 6.

Solving linear equations

Equations 1

Is x = 5 a solution to these equations?

(a) x + 1 = 4

(b) 3x − 4 = 1 + 2x

(c) 2(x − 1) − (13 − x) = 0

Equations 3

Show (without solving the equation) that x = −2 is a solution to the equation 3 − 5x = 15 + x. Now solve the equation.

Solve the following equations.

(a) 2a = 22

(b) a + 3 = 8

(c)

(d) 2k = 7 + k

(e) y = 12 − 2y

Equations 5

13

x

Solve the following equations.

(a) B + 13 = 7 − 5B

(b) x − 1 = 2(x + 1)

(c)

(d)

(e)

Equations 8

31

6

x

3( 7) 142

yy

51

3 6

a a

Finding an unknown number

Problems 1 A table has been reduced in price by 25%. It now costs £270. Express this information as an equation and then solve the equation to find how much the table cost before the reduction.How would your equation need to change if the price reduction was 20%?

Problems 3

Sara is five times the age she was 28 years ago. How old is Sara?

Problems 5

In four years time, Tom will be twice the age he was 11 years ago. How old is Tom?

Problems 7

A man is five times as old as his son. In two years time, he will only be four times as old as his son. How old is the son now?

Problems 9 Bob has 2 children, one two years older than the other. At present Bob is twice as old as the sum of his children’s ages. In 14 years time, Bob’s age will be exactly the sum of the ages of his two children. How old are the children now?

Problems 11

A number is divided by 7 and then 6 is added to the result giving 15. What number did you start with?

Problems 13

The sum of three consecutive integers is 351. Find the numbers.

Problems 15

Nicholas buys two books for his nieces, Amy and Becky, for Christmas. Becky’s book costs £5 more than Amy’s and together the bill came to £21. How much were the books?

Problems 17

Great Uncle Theo is planning some donations to charity. He has £1000 which he wants to divide between the Cats’ Home, the Dogs’ Home and the sanctuary for parrots. He wants to give the Dogs’ Home twice as much as the Cats’ Home. He also wants to give £100 more to the sanctuary for parrots than to the Cats’ Home. How much does he give to each?

Problems 19

There are four people travelling in a car. Bill and Ben are the same age, Molly is twice as old as Bill and Pete is twenty years older than Molly. The sum of their ages is 86. How old is Molly?

Problems 19

There are four people travelling in a car. Bill and Ben are the same age, Molly is twice as old as Bill and Pete is twenty years older than Molly. The sum of their ages is 86. How old is Molly?

Magic squares 1

A magic square has the property that the numbers in each column, each row and each diagonal add to the same total. Substitute a = 5, b = 3 and c = −1 in the square below and check that the result is a magic square.

a + c a + b − c a − b

a − b − c a a + b + c

a + b a − b + c a − c

Now use algebra to explain why the property works for all values of a, b and c.