# Maths

Maths, mainly algebra

- Created by: Jacinta
- Created on: 18-05-10 18:08

## Algebra; Trial and Improvment

An equation such as x³ + x = 50 does not have an exact solution: the answer is a decimal number. Find the answer correct to 1 decimal place. We are looking for a number which, when you cube it and add the number itself, you get the answer 50.

First guess: x = 3 3 × 3 × 3 + 3 = 27 + 3 = 30 - too small

- Second guess x = 4 4 × 4 × 4 + 4 = 64 + 4 = 68 - too big

- We now know that the answer lies between 3 and 4.
- Third guess: x = 3.5 3.5 × 3.5 × 3.5 + 3.5 = 42.875 + 3.5 = 46.375 - too small

- We now know that the answer lies between 3.5 and 4.
- Fourth guess x = 3.6 3.6 × 3.6 × 3.6 + 3.6 = 46.656 + 3.6 = 50.256 - too big

We now know that the answer lies between 3.5 and 3.6. But it must be closer to 3.6, so the answer is x = 3.6 correct to 1 decimal place.

## Terms

We can often simplify algebraic expressions by '**collecting like terms**'.

Look at the expression .

There are four **terms** and .

Two of the terms involve , and two involve .

We can **re-order** the terms in the expression so that the x terms are together and the y terms are together:

.

Now we can combine the x terms and combine the y terms to get:

.

So, when simplified, becomes .

## Algebra Simultaneous Equations

Equation 1: 2x + y = 7

- Equation 2: 3x - y = 8

Add the two equations to eliminate the **y**s:

- 2x + y = 7
- 3x - y = 8
- ------------
- 5x = 15
- x = 3

- Now you can put x = 3 in either of the equations.
- Substitute x = 3 into the equation 2x + y = 7:
- 6 + y = 7
- y = 1

So the answers are **x = 3** and **y = 1...Sometimes you will need to multiply one of the equations before you can add or subtract.**

## Still Simultaneous Equations

Solve the simultaneous equations:

Equation 1: y - 2x = 1

Equation 2: 2y - 3x = 5

Rearranging Equation 1, we get y = 1 + 2x

We can replace the ‘y’ in equation 2 by substituting it with 1 + 2x

Equation 2 becomes: 2(1 + 2x) - 3x = 5

2 + 4x - 3x = 5

2 + x = 5

x = 3

Substituting x = 3 into Equation 1 gives us y - 6 = 1, so y = 7.

## Simple Equations

Solve the equation **2a + 3 = 7**

This means we need to find the value of *a*. The answer is **a = 2**

**Write down the expression: 2x +3 = 7****Then choose a value for****a**that looks about right and work out the equation. Try 3.- a = 3, so 2
**×**3 + 3 = 9 -
Using 3 to represent

**a**makes the calculation more than 7, so choose a smaller number for**a**. - Try a = 2, then 2
- 2
**×**2 + 3 = 7

Which gives the right answer. So a = 2

Be systematic in your approach:

- choose a number, work it out, then move the number up or down

## Simple Equations

The best way to solve an equation is by using 'inverses', or undoing what the equation is doing.

- Adding and subtracting are the inverse (or opposite) of each other.
- Multiplying and dividing are the inverse of each other.

- First write down the expression:
- 2a + 3 = 7
- Then undo the + 3 by subtracting 3. Remember, you need to do it to BOTH sides!
- 2a + 3 - 3 = 7 - 3
- so 2a = 4
- Undo the multiply by 2 by dividing by 2 - again on both sides:
- 2a ÷ 2 = 4 ÷ 2

The answer is: **a = 2**

## Algebra- Brackets

To remove brackets, we multiply them out.

Look at the expression .

This expression means everything inside the brackets is **multiplied by****.**

Therefore becomes when the brackets are removed.

## Example

Multiply out the brackets in

This is because and .

## Inequalities

Inequalities are expressions which indicate a variable

## Symbols and their meaning

SymbolMeaning < **is less than**, so 2 < 5 is a true statement. > **is more than**, so 6 > 4 is a true statement.

**is less than or equal to**

so 2 5 is true

and so is 2 2.

is more than or equal to,

so 6 4 is true,

and so is 6 6.

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