Course Content
Chapter 01 – Operations on Sets
The set operations are performed on two or more sets to obtain a combination of elements as per the operation performed on them. In a set theory, there are three major types of operations performed on sets, such as: Union of sets (∪) The intersection of sets (∩) Difference of sets ( – ) In this lesson we will discuss these operations along with their Venn diagram and will learn to verify the following laws: Commutative, Associative, Distributive, and De-Morgans' law.
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Subsets and Super Sets
03:10
Venn Diagrams
03:32
Operations on Set
00:00
Union of Sets
16:15
Intersection of Sets
17:12
Difference of Sets
13:13
Complement of Sets
19:42
Venn Diagrams (Three or More Sets)
03:22
Commutative Law of Union and Intersection
02:53
Associative Law of Union and Intersection
05:01
Distributive Law over Union and Intersection
04:13
De Morgan’s Law
00:00
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Chapter 02 – Real Numbers
All real numbers follow three main rules: they can be measured, valued, and manipulated. Learn about various types of real numbers, like whole numbers, rational numbers, and irrational numbers, and explore their properties. In this chapter, we will learn about Squares and cubes of real numbers and find their roots.
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Irrational Numbers
03:14
Squares
03:04
Square Roots
03:49
Cubes and Cube Roots
00:00
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Chapter 03 – Number System
The number system or the numeral system is the system of naming or representing numbers. There are different types of number systems in Mathematics like decimal number system, binary number system, octal number system, and hexadecimal number system. In this chapter, we will learn different types and conversion procedures with many number systems.
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Number System
08:13
Conversion – Base 2
00:00
Addition in Binary System
03:02
Subtraction in Binary System
02:32
Multiplication in Binary System
02:43
Conversion – Base 5
00:00
Addition in Base 5
01:55
Subtraction in Base 5
01:38
Multiplication in Base 5
04:17
Conversion – Base 8
08:19
Addition & Subtraction in Base 8
06:07
Multiplication in Base 8
04:39
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Chapter 04 – Financial Arithmetic
Financial mathematics describes the application of mathematics and mathematical modeling to solve financial problems. In this chapter, we will learn about partnership, banking, conversion of currencies, profit/markup, percentage, and income tax.
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Compound Proportion
05:10
Partnership
07:18
Inheritance
09:54
Conversion of Currencies
03:13
Profit or Markup
07:01
Percentage
08:31
Taxes
03:46
Income Tax
02:16
Banking
06:17
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Chapter 05 – Polynomials
In algebra, a polynomial equation contains coefficients, exponents, and variables. Learn about forming polynomial equations. In this chapter, we will study the definition and the three restrictions of polynomials, we'll tackle polynomial equations and learn to perform operations on polynomials, and learn to avoid common mistakes.
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Algebraic Expression
00:00
Polynomials
00:00
Addition and Subtraction of Polynomial
00:00
Multiplication of Polynomials
02:43
Division of Polynomial
00:00
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Chapter 06 – Factorization, Simultaneous Equations
In algebra, factoring is a technique to simplify an expression by reversing the multiplication process. Simultaneous Equations are a set of two or more algebraic equations that share variables and are solved simultaneously. In this chapter, we will learn about factoring by grouping, review the three steps, explore splitting the middle term, and work examples to practice verification and what simultaneous equations are with examples. Find out how to solve the equations using the methods of elimination, graphing, and substitution.
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Algebraic Formulas
06:35
Stimulation – Area Model Algebra
00:00
Factorization
00:00
Manipulation of Algebraic Expression
00:00
Stimulation – Equality Explorer
00:00
Linear Equations in One Variable
00:00
Simultaneous Linear Equations
00:00
Stimulation – Equality Explorer (Two Variables)
00:00
Elimination
00:00
Let's Try
Chapter 07 – Fundamentals of Geometry
Geometry is the study of different types of shapes, figures, and sizes. It is important to know and understand some basic concepts. We will learn about some of the most fundamental concepts in geometry, including lines, polygons, and circles.
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Parallel Lines
05:36
Polygons
00:00
Circles
00:00
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Chapter 08 – Practical Geometry
Geometric construction offers the ability to create accurate drawings and models without the use of numbers. In this chapter, we will discover the methods and tools that will aid in solving math problems as well as constructing quadrilaterals and right-angled triangles.
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Construction of Quadrilaterals
11:16
Construction of a Right Angled Triangle
00:00
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Chapter 09 – Areas and Volumes
The volume and surface area of a sphere can be calculated when the sphere's radius is given. In this chapter, we will learn about the shape sphere and its radius, and understand how to calculate the volume and surface area of a sphere through some practice problems. Also, we will learn to use and apply Pythagoras' theorem and Herons' formula.
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Pythagoras Theorem
00:00
Heron’s Formula
00:00
Surface Area, and Volume of a Sphere
00:00
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Math Lab
Chapter 10 – Demonstrative Geometry
Demonstrative geometry is a branch of mathematics that is used to demonstrate the truth of mathematical statements concerning geometric figures. In this chapter, we will learn about theorems on geometry that are proved through logical reasoning.
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Reasoning
03:34
Axioms
00:00
Postulate
05:01
Theorems
00:00
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Chapter 11 – Trigonometry
Sine and cosine are basic trigonometric functions used to solve the angles and sides of triangles. In this chapter, we will review trigonometry concepts and learn about the mnemonic used for sine, cosine, and tangent functions.
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What is Trigonometry?
00:00
Visualization of Trigonometric Ratios
02:55
Trigonometric Ratios of Complementary Angles
05:21
Simulation – Trig Tour
00:00
Angle of Elevation and Angle of Depression
04:22
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Chapter 12 – Information Handling
Frequency distribution, in statistics, is a graph or data set organized to show the frequency of occurrence of each possible outcome of a repeatable event observed many times. Measures of central tendency describe how data sets are clustered in a central value. In this chapter, we will learn to construct the frequency distribution table, and learn more about three measures of central tendency, its importance, and various examples.
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Frequency Distribition
01:03
Bar Graph
07:49
Histogram
00:00
Pie Chart
04:29
Measures of Central Tendency
00:00
Mean, Median, and Mode
00:00
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Grade 8 – Mathematics
About Lesson

Simultaneous Equations

Simultaneous equations are two or more algebraic equations that share variables e.g. x and y. They are called simultaneous equations because the equations are solved at the same time.

Solving Simultaneous Linear Equations | PBS LearningMedia

For example, below are some simultaneous equations:

2x + 4y = 14,         4x − 4y = 4

6a + b = 18, 4a + b = 14

3h + 2i = 8,          2h + 5i = −2

Each of these equations on their own could have infinite possible solutions. However when we have at least as many equations as variables we may be able to solve them using methods for solving simultaneous equations.

We can consider each equation as a function which, when displayed graphically, may intersect at a specific point. This point of intersection gives the solution to the simultaneous equations. E.g.

−3x + y = 2

Simultaneous equations graph

When we draw the graphs of these two equations, we can see that they intersect at (1,5).

So the solutions to the simultaneous equations in this instance are:

x = 1 and y = 5

Solving simultaneous equations

When solving simultaneous equations you will need different methods depending on what sort of simultaneous equations you are dealing with.

A linear equation contains terms that are raised to a power that is no higher than one. E.g. 

Linear simultaneous equations are usually solved by what’s called the elimination method (although the substitution method is also an option for you).

What is substitution?

Substitution means replacing the variables (letters) in an algebraic expression with their numerical values. We can then work out the total value of the expression.

We can substitute values into formulae to help us work out many different things. Examples range from the formula for the area of a triangle:

How to substitute a value into an algebraic expression

In order to substitute into an algebraic expression:

  1. Rewrite the expression substituting each letter with its given numerical value.
  2. Calculate the total value of the expression. Remember that you must apply BIDMAS.

Substitution is a useful method to use when one or more of the equations is not linear. Substitution works by making one of the variables the subject of one of the equations and then substituting this into the other equation.

For example, let’s solve the simultaneous equations

Step 1: Make one of the variables the subject of one of the equations.

Step 2: Substitute this into the other equation.

Step 3: Solve the equation.

Step 4: Substitute into one of the original equations to find the other value.

Step 5: Check your answer by substituting into the other original equation.

These both equal 1, so our solutions are correct.

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