Mr. J Gallagher

Algebra - Worksheet

Junior Cert - Higher Level

Syllabus

You should be able to:

Explain / understand the following key words

  • Variable
  • Term
  • Coefficient
  • Expression
  • Factors
  • Solve an equation
  • Linear equation
  • Simultaneous equations
  • Factors / Roots
  • Inequalities
  • Lowest Common Denominator (L.C.D.)

Calculate value of expressions by using substitution

Add / subtract like terms

Multiply terms (with brackets)

Simplify algebraic fractions

Factorising - all types

Solve quadratic equations using formula

Solve linear equations - all types

Solve inequalities – normal & compound inequalities and graph on number line

Solve simultaneous equations

Problem solving in algebra

Add and subtract the like terms

Simplify the following expressions

a)16a – 4c + 10b + 6a – 7b + 3c

b)-14x + 11y + 4x – 13y

Evaluating Expressions
Evaluate the following expressions when x = 2, y = -3

a)x2 – 4xy

b)6xy + 3x2y

c)x2 – y2

2011 Exam Question (5 Marks)

2014 Exam Question (10 Marks)

2012 Exam Question (15 Marks)

Multiplying Brackets

Simplify the following expressions

a)4(3x - 4) + 3(-2x + 5)

b)-3(2x – 6y) – 4(x – y)

c)7(2a + 3b) +6(4a – 2b) - 4(3b – 2a)

d)(3x - 4) (-2x + 5)

e)(2x – 6) (x – 1)

f)(2a + 3) (3b – 2)

2014 Exam Question (5 Marks)

2015 Exam Question (5 Marks)

2012 Exam Question (10 Marks)

Linear Equations

Solve the following equations

a)x + 2 = 0

b)4x + 3 = 15

c)3y – 6 = 18

d)3x – 1 = 2x + 11

Linear equations with brackets - Solve the following equations

a)3(x + 2) = 0

b)-3(4x + 3) = 15

c)3(2y – 6) = 18

d)4(x – 2) = 3(2x + 4)

e)3(x – 2) = 7(x + 5) – 13

f)2(2x + 1) – 3 (x – 1) = 9

Check your answers using substitution!

Simplifying Fractions

Simplify the following expression

a), hence solve

b), hence solve

c), hence solve

d), hence solve

Check your answers using substitution!

2011 Exam Question (10 Marks)

2011 Exam Question (15 Marks)

2012 Exam Question (10 Marks)

2015 Exam Question (10 Marks)

Simultaneous Equations

Linear Simultaneous Equations – Solve the following for x & y using both the Algebraic and Graphical method. Also check your answer using substitution.

a)x – y = 1

2x + y = 11

b)3x + 2y = 8

2x – 2y = 2

c)2x + 3y = 8

5x + 3y = 11

d)x + 2y = 8

2x + 3y = 14

e)x – 2y = 9

3x + 7y = 1

f)3x + 4y = 23

y = 2x + 3y

2014 Exam Question (15 Marks)

Inequalities

Linear – Solve the following inequality and graph your solution on a number line

a)2x – 3 5, x R

b)3x – 1 > 8, x N

c)4x + 3 < 3x + 10, x R

d)6(x + 4) > 2(x – 3)x R

2011 Exam Question (10 Marks)

2015 Exam Question (10 Marks)

Compound - Solve the following inequality and graph your solution on a number line

a)-5 < 3x + 1 7, x R

b)– 3 2x – 3 < 7, x R

c)– 9 < 4x + 3 15,x R

2014 Exam Question
(10 Marks)

Factorising

Common Factors - Factorise the following expressions

a)x2 + 2x

b)2y2 + 4y

c)4xy – 12x2

d)3x2y – 15xy

Difference of two squares - Factorise the following expressions

a)x2 – 49

b)y2 – 144

c)4x2 – 25

d)36y2 – 16x2

2011 Exam Question (5 Marks)

Common Terms & Difference of Two Squares

a)x2 – 1

b)81y2 – 16a2 b2

c)3x2y2 – 12

d)32x2 – 18y2

e)125x2 – 5

Grouping factors - Factorise the following expressions

a)ax + ay + bx + by

b)3ax – 2ay + 3bx – 2by

c)4x – 4y + abx – aby

2011 Exam Question (5 Marks)

2015 Exam Question (5 Marks)

Factorising quadratics - Factorise the following expressions

a)x2 – x – 20

b)x2 + 3x – 10

c)4x2 + 4x + 1

d)14x2 + 3x – 2

2011 Exam Question (5 Marks)

Quadratic equations - Solve the following equations

a)x2 – x – 20 = 0

b)x2 + 3x – 10 = 0

c)4x2 + 4x + 1 = 0

d)14x2 + 3x – 2 = 0

e)x2 + 8x + 12

f)x2 – 12x + 27

g)9x2 + 25x – 6

h)3x2 + 11x – 20

i)12x2 – 11x – 5 = 0

Solve each of the following quadratic equations. Round to 2 decimal places.

a)3x2 – 6x + 2=0

b)10x2 + 17x + 7=0

c)x2 + 12x + 20 = 0

d)x2 - 9x + 18 = 0

e)12x2 – 11x – 5 = 0

f)4x2 – 11x + 6 = 0

g)9x2 + 25x – 6 = 0

Form a quadratic equation given the two roots below:

a)-5, -4

b)3, 2

c)3, -4

d)-6, 1

Simplify the following expressions (Division)

a)

b)

2012 Exam Question (15 Marks)

2012 Exam Question (25 Marks)

2014 Exam Question (20 Marks)

2015 Exam Question (20 Marks)

Equations with x as index

Solve for x

a)3x = 27

b)2x+1 = 16

c)32x+1 = 243

d)9x+1 =

e)32x+1 = 3

f)22x-2 =

g)49x = 72+x

Real Life Examples

  1. Brendan thinks of a number, adds three and the answer is fifteen. Represent this statement as an equation. Solve the equation and check your answer.
  1. Ryan thinks of a number then subtracts five and the answer is ten. Represent this statement as an equation. Solve the equation and then check your answer.
  1. A farmer has a number of cows and he plans to double that number next year, when he will have twenty-four. Represent this statement as an equation. Solve the equation and check your answer.
  1. A new student enters class and the class now has twenty-five students. Represent this statement as an equation. Solve the equation and check your answer.
  1. The temperature increases by eighteen degrees and the temperature is now fifteen degrees. Represent this statement as an equation. Solve the equation and check your answer.
  1. A farmer doubles the amount of cows he has and then buys a further three cows. He now has twenty-nine. Represent this statement as an equation. How many did he originally have?
  1. Emma and her twin brother will have a total age of forty-two in five years time. Represent this statement as an equation. How old are they at the moment?
  1. Mark had some cookies. He gave half of them to his friend John. He then divided his remaining cookies evenly between his other three friends each of whom got four cookies. How many had he originally?

2011 Exam Question (20 Marks)

2014 Exam Question (20 Marks)

Simultaneous Equations

  1. The combined cost of a television and a DVD player is 1460euro. The television costs 330euro more than the DVD player.
  1. Use two equations in x and y to represent the situation
  2. Hence, find the cost of the television and the cost of the DVD player.
  1. Two numbers have a difference of 13. Twice the bigger number added to 19 times the smaller number makes 110.
  1. If x is the bigger number and y is the smaller number, write down two equations in x and y.
  2. Hence, solve for the two numbers.
  1. Let the cost of a meal for an adult be x euro and the cost of a meal for a child be y euro. The cost of a meal for three adults and two children amounts to 125euro. The cost of a meal for two adults and three children amounts to 115euro.
  1. Write down two equations in x and y to represent this information.
  2. Solve these equations to find the cost of an adults meal and the cost of a child’s meal.
  1. A builders’ supplier sells two types of copper pipes. One has a narrow diameter and costs x euro per length. The other has a wider diameter and costs y euro per length. Tony buys 14 length of the narrow pipes and 10 lengths of the wider pipes at a cost of 555euro. Gerry buys 12 lengths of the narrow pipes and 5 lengths of the wider pipes at a cost of 390euro.
  1. Write two equations to represent the above information.
  2. Solve these equations for find the cost of the narrow and wider pipes.

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