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How to draw a HISTOGRAM using FREQUENCY DENSITY

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How to draw a HISTOGRAM using FREQUENCY DENSITY

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Averages from tables - Analysing data - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

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Learn about and revise how to find the mean, median and mode averages from data with this BBC Bitesize GCSE Maths Edexcel study guide.

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Time series - Representing data - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

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Learn about and revise how to display data on various charts and diagrams with this BBC Bitesize GCSE Maths Edexcel study guide.

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SOLVING LINEAR EQUATIONS

Recall the following:

A linear equation is a polynomial of degree 1.

In order to solve for the unknown variable, you must isolate the variable

In the order of operations, multiplication and division are completed before addition and subtraction.

Problem 1.1a:
x + 7 = 3

Answer: x = - 4

Solution: Subtract 7 from both sides of the equation.

which gives:

x = - 4

Check: Substitute -4 for x in the original equation. If, after the substitution, the left side equals the right side, your answer is correct.( -4 ) + 7 = 3 Your answer is correct.

Problem 1.1b:
x - 3 = 4

Answer: x = 7

Solution: Add 3 to both sides of the equation

which gives:

x = 7

Check: Substitute 7 forx in the original equation. If, after the substitution, the left side equals the right side, your answer is correct.( 7 ) - 3 = 4 . Your answer is correct.

Problem 1.1c:
x - 17 = 3

Answer: x = 20

Solution: Add 17 to both sides of the equation.

which gives:

x = 20

Check: Substitute 20 for x in the original equation. If, after the substitution, the left side equals the right side, your answer is correct. (20) - 17 = 3 . Your answer is correct.

Problem 1.1d:
x + 5 = 8

Answer: x = 3

Solution: Subtract 5 from both sides of the equation.

which gives:

x = 3

Check: Substitute 3 for x in the original equation. If, after the substitution, the left side equals the right side, your answer is correct. (3) + 5 = 8. Your answer is correct.

Problem 1.1e:

Answer:

Solution: Add to both sides of the equation.

which gives:

Add the fractions.

Check: Substitute for x in the original equation. If, after the substitution, the left side equals the right side, your answer is correct. .

Your answer is correct.

If you would like to go back to the equation Table of contents, click on Contents.

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Recall the following: A linear equation is a polynomial of degree 1. In order to solve for the unknown variable, you must isolate the variable In the order of operations, multiplication and division are completed before addition and subtraction. Problem 1.1a: x + 7 = 3 Answer: x = - 4 Solution: Subtract 7 from both sides of the equation.

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Algebraic Expressions With Example Problems and Interactive Exercises

Use the following examples and interactive exercises to learn about Writing Algebraic Expressions.

Problem: Ms. Jensen likes to divide her class into groups of 2. Use mathematical symbols to represent all the students in her class.

Solution: Let g represent the number of groups in Ms. Jensen's class.

Then 2 · g, or 2g can represent "g groups of 2 students".

In the problem above, the variable g represents the number of groups in Ms. Jensen's class. A variable is a symbol used to represent a number in an expression or an equation. The value of this number can vary (change). Let's look at an example in which we use a variable.

Example 1: Write each phrase as a mathematical expression.

The expression 9 + 8 represents a single number (17). This expression is a numerical expression, (also called an arithmetic expression). The expression 9 + x represents a value that can change. If x is 2, then the expression 9 + x has a value of 11. If x is 6, then the expression has a value of 15. So 9 + x is an algebraic expression. In the next few examples, we will be working solely with algebraic expressions.

Example 2: Write each phrase as an algebraic expression.

Phrase Expression

nine increased by a number x

9 + x

fourteen decreased by a number p

14 - p

seven less than a number t

t - 7

the product of 9 and a number n

9 · n or 9n

thirty-two divided by a number y

32 ÷ y or

In Example 2, each algebraic expression consisted of one number, one operation and one variable. Let's look at an example in which the expression consists of more than one number and/or operation.

Example 3: Write each phrase as an algebraic expression using the variable n.

Phrase Expression

five more than twice a number

2n + 5

the product of a number and 6

6n

seven divided by twice a number

7 ÷ 2n or

three times a number decreased by 11

3n - 11

Example 4: A small company has $1000 to distribute to its employees as a bonus. How much money will each employee get?

Solution: Let e represent the number of employees in the company. The amount of money each employee will get is represented by the following algebraic expression:

Example 5: An electrician charges $45 per hour and spends $20 a day on gasoline. Write an algebraic expression to represent his earnings for one day.

Solution: Let x represent the number of hours the electrician works in one day. The electrician's earnings can be represented by the following algebraic expression:

Solution: 45x - 20

Summary: A variable is a symbol used to represent a number in an expression or an equation. The value of this number can change. An algebraic expression is a mathematical expression that consists of variables, numbers and operations. The value of this expression can change.

Exercises

Directions: Choose the algebraic expression that correctly represents the phrase provided. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.

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Learn How To Write and Understand an Algebraic Expression

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Use the following examples and interactive exercises to learn about Writing Algebraic Expressions. Problem: Ms. Jensen likes to divide her class into groups of 2. Use mathematical symbols to represent all the students in her class. Solution: Let g represent the number of groups in Ms. Jensen's class.

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Standard form is a way of writing down very large or very small numbers easily. 10 ³ = 1000, so 4 × 10 ³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative. The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

This video explains standard form:
Example
Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 10 ¹³

It's 10 ¹³ because the decimal point has been moved 13 places to the left to get the number to be 8.19
Example
Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10⁻⁶

It's 10⁻⁶ because the decimal point has been moved 6 places to the right to get the number to be 1.2

On a calculator, you usually enter a number in standard form as follows: Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.
Manipulation in Standard Form
This is best explained with an example:
Example
The number p written in standard form is 8 × 10 ⁵

The number q written in standard form is 5 × 10⁻²

Calculate p × q. Give your answer in standard form.

Multiply the two first bits of the numbers together and the two second bits together:

The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.
Example
Calculate p ÷ q.

Give your answer in standard form.

This time, divide the two first bits of the standard forms. Divide the two second bits. (8 ÷ 5) × (10 ⁵ ÷ 10⁻²) = 1.6 × 10 ⁷

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Standard Form - Mathematics GCSE Revision

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Standard form is a way of writing down very large or very small numbers easily. 10 3 = 1000, so 4 × 10 3 = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

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Adding Fractions with Unlike Denominators

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Histograms - Representing data - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

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Learn about and revise how to display data on various charts and diagrams with this BBC Bitesize GCSE Maths Edexcel study guide.