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Maths, mainly algebra

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  • 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.

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Terms

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

Look at the expression 2x + 5y + x - 3y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_fbac4a75_5601_37c4_897c_ab7459ababd4.gif).

There are four terms2x, 5y, x (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_151f711e_5522_3075_b773_b70656656066.gif) and -3y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_a123f20e_bd12_3945_bc9d_48a2ee657ed1.gif).

Two of the terms involve x (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_5ee7f6b3_792f_3d58_9a9e_05c4946b176e.gif), and two involve y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_84baebd8_92ff_36d1_88dc_4c4cf1fe3b41.gif).

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

2x + x + 5y - 3y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_81b30ace_d436_3f56_bee5_636b77da42bf.gif).

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

3x + 2y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_f27a2e87_2309_3302_9a57_2b7a7b820370.gif).

So, when simplified, 2x + 5y + x - 3y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_fbac4a75_5601_37c4_897c_ab7459ababd4.gif) becomes 3x + 2y (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_f27a2e87_2309_3302_9a57_2b7a7b820370.gif).

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Algebra Simultaneous Equations

Equation 1: 2x + y = 7

  • Equation 2: 3x - y = 8

Add the two equations to eliminate the ys:

  • 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.

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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.

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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:

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

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Algebra- Brackets

To remove brackets, we multiply them out.

Look at the expression 4(y - 5) (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_82324539_813b_371d_9e94_181e3a5c3b5f.gif).

This expression means everything inside the brackets is multiplied by4 (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_103d07c2_5a58_31d7_9eac_d15d9a856f08.gif).

4(y - 5) (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_82324539_813b_371d_9e94_181e3a5c3b5f.gif)

= 4 \times y - 4 \times 5 (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_61446ec6_0121_3e9d_aeb0_1dc9ff295f7e.gif)

= 4y - 20 (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_81b2a7fb_b5ac_3dd9_a072_44e13c5c2918.gif)

Therefore 4(y - 5) (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_82324539_813b_371d_9e94_181e3a5c3b5f.gif) becomes 4y - 20 (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_6c3ae466_8c98_3675_b541_b40cb4834623.gif) when the brackets are removed.

Example

Multiply out the brackets in 3(2a + 1) (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_3245a14c_390e_360f_bd67_2f4526e12541.gif)

3(2a + 1) = 6a + 3 (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_fd3419e8_6b25_3aa7_bead_99866708b10d.gif)

This is because 3 \times 2a = 6a (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_1f30808f_d2cc_30a2_a41a_5d5d3f63bdf7.gif) and 3 \times 1 = 3 (http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/relationships/images/tex_61dbfb35_f5df_3a46_90d6_ab8993d1cbeb.gif).

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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.

less than or equal to (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/lessthaneq.png)

is less than or equal to

so 2 less than or equal to (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/lessthaneq.png) 5 is true

and so is 2 less than or equal to (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/lessthaneq.png)2.

greater than or equal to (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/greaterthaneq.png)

is more than or equal to,

so 6 greater than or equal to (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/greaterthaneq.png) 4 is true,

and so is 6 greater than or equal to (http://www.bbc.co.uk/schools/gcsebitesize/maths/images/greaterthaneq.png) 6.

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