Maths - Formulae and Equations

Maths - Formulae and Equations

1. Using Formula

   • A taxi firm charges £0.50 per miles plus a fixed rate of £2.00
     • It costs £2 + £0.50 to travel 1 mile, It costs £2 + 2 x £0.50 to travel 2 miles ...
       • So travelling 'n' miles will cost £2 + n x £0.50
         • The formula is COST = £2 + (n x £0.50)
           • Substitution
             • What is the cost of hiring the taxi for 16 miles?
               • C = £2 + (16 x £0.50)
                 • C = £2 + £8
                   • C = £10
   • A rectangle has a width of x and a length of 2x
     • Perimeter = x + x + 2x + 2x
       • P = 6x

2. Re-arranging Symbols

   • Collecting like terms
     • To simplify an expression, we collect like terms
       • 4x + 5x - 2 - 2x + 7
         • The x terms can be collected together and the numbers can be collected together
           • So 4x + 5x - 2x = 7x and 7 - 2 = 5
             • This simplifies to 7x + 5
       • x + 5 + 3x - 7 + 9x + 3 - 4x
         • So x + 3x + 9x -4x = 9x and 5 + 3 - 7 =1
           • This simplifies to 9x + 1
   • Different Terms
     • You may have to simplify an equation with many different terms and letters
       • 5a + 3b - 3a - 5c + 4b
         • Collect like terms
           • 5a - 3a +3b +4b - 5c
             • Simplify all terms
               • 2a + 7b - 5c
   • Multiplying out brackets
     • 3(4x - 7)
       • First Multiply: 3 x 4x = 12x
         • Then multiply: 3 x -7 = -21
           • Therefore: 3(4x - 7) = 12x - 21
     • Remember whether the numbers are negative or positive
     • Multiplying out two brackets
       • (x+4)(x+3)
         • FOIL
           • x2 + 7x + 12
       • Brackets and Powers
         • (a-5) squared
           • (a-5)(a-5)
             • a2 - 10a + 25

3. Changing the subject of a formula

   • Arrange the formula C=2pi r to make r the subject
     • Divide both sides by 2pi
       • r = C/ 2pi
   • Rearrange the formula V = 4/3 pi r (3)
     • Multiply by 3
       • 3V = 4pi r (3)
         • Divide by 4 pi
           • 3V/ 4 pi = r(3)
             • Take the cube root of both sides
               • r = cube root of 3V / 4pi

4. Simultaneous Equations

   • 2x + y = 7
     • 3x - y = 8
       • Add the equations
         • 5x = 15 so...
           • x = 3
             • Substitute in...
               • (2 x 3) + y = 7
                 • y = 7-6
                   • x = 3, y = 1
   • If the item you want to remove is two positives or two negatives you subtract, if they are one positive and one negative you add

5. Equations with Fractions

   • x/2 - 4 = 3
     • +4 so x/2 = 7
       • x2 so x = 14

6. Trial and Improvement

   • Always see to how many d.p the answer needs to be
     • Remember to get as close to the final answer

7. Index Notation

   • Powers
     • a squared = a x a
       • b cubed = b x b x b
         • 4d squared = 4 x d x d
   • Index Laws
     • When multiplying you add the indices, when subtracting you divide the indices
       • p3 x p7 = p10
         • 4s3 x 3s2 = 12s5

8. Solving Equations

   • Using inverses
     • x - 6 = 9
       • x = 9+6
         • x = 15
   • Unknowns on both side
     • 3b + 4 = b + 12
       • -b so 2b + 4 = 12
         • -4 so 2b = 8
           • /2 so b = 4
   • Equations with brackets
     • 3(b+2) = 15
       • 3 x b + 3 x 2 = 15
         • 3b +6 = 15
           • 3b = 9
             • b = 3