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Coursework Notes - Algebra


Simple Equations


equation rules   substitution   changing the subject   creating a formula


Simple Equations are equations with one variable only(usually denoted by 'x').


The 'Golden Rule' - whatever operations are performed, they must operate equally on both sides of the equation.



general method for solving equations:

  • remove fractions by multiplying both sides of the equation by a common denominator

  • expand any brackets

  • take 'x'(or variable) terms to the left

  • take number terms and unwanted variable terms, to the right

  • collect terms on each side

  • if the variable(x) on the left is multiplied by a number, divide both sides by that number(the number cancels on the left)

  • cancel divided terms where necessary


a practical note:


To move a term from one side of an equation to the other, change its sign.


In effect a term of the same value but opposite in sign is added to each side of the equation. This cancels on one side giving the appearance of the original term having moved across the equation with its sign changed.


changing sign across an equation


In our example -3 is added to each side of the equation. This appears to have the effect of moving the +3 across the equals sign and changing it to -3 .



Example #1


equations eg#1



Example #2


simple equation eg#2



Example #3


equation eg. #3



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Substituting values into an equation - Simply write the equation again with the letters replaced by the numbers they represent. Use brackets to avoid arithmetic errors.



Example #1


equation substitution eg#1



Example #2

substitution #2



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Changing the subject of an equation - Simply follow these simple rules:


  • move all terms containing the subject to the LHS of the equation either by moving them across the = sign & changing their sign, or flipping the equation over horizontally, so that the LHS is on the right & vice versa.

  • factorise the terms containing the subject

  • divide both sides of the equation by the contents of any brackets.

  • remove any unwanted negative signs on the left by multiplying both sides by -1

  • remove fractions by multiplying both sides by the denominator(lower number of any fraction)



Example #1 make 'b' the subject of the equation


changing the subject eg#1



Example #2 make 'd' the subject of the equation


changing the subject eg #2



Example #3 make 'c' the subject of the equation


changing the subject eg #3



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Creating a formula - Often when creating a formula some basic knowledge is required:


e.g. area of a triangle = half base time height,


area rectangle = width x length etc.



Example #1 A plant of original length letter l nought cm. grows at a rate of a cm. per day. If days are represented by the letter d , write an expression for the height h of the plant in cm.


create formula #3



Example #2 A box has height a, length b and width c . Write an expression to represent the surface area A, of the box.


create formula eg2



Example #3 A square box of side L contains a sphere, where the radius of the sphere is equal to half the side of the box. Write an expression for the volume in the box not taken up by the sphere.


create formula #5



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