Angol Matematika

Equations

Def: an equation is a mathematical statement that asserts (kijelent) the equality of two expressions.

Equations express relationships between given quantities, the knowns and quantities to be determined, the unknowns.

By conventions, unknowns are denoted by letters at the end of the alphabet (x, y,..) while knowns are denoted by letters at the beginning of the alphabet (a,b,…) or numbers.

To deal with an equation we have to give the fundamental set (the domain) of the equation.

If the fundamental set is not given, then the largest set for which the expressions of the equation are defined (having meaning) is regarded as the domain.

The solutions of an equation are the set of all elements of the fundamental set which satisfy the equation. If we can find all these elements, then the equation is solved. These numbers are said to be the solutions of the equation or the roots of the equations.

Methods to find the solution:

-graphical method: represent the functions on the left and right-hand sides in a coordinate system. Read off the x coordinate(s) of the common point(s) of the two graphs.

-by considering products: it is useful to transform equations of higher degree into products that are equal to zero.

-rearrangement: !! if we don’t change the fundamental set of equation when we perform transformations, then the new equation will be satisfied by the same number(s) as the original one.

Steps to solve an equation:

1. set the fundamental set of the equation

2. remove the brackets wherever it’s possible

3. to eliminate fractional expressions, multiply the equation by LCM of the denominators.

4. collect the like terms on both sides of the equation

5. collect terms which contain the variable on one side of the equation and move numbers to the other

6. reduce the equation, express the variable

7. CHECK!

Only transformations which do not yield (eredményez) new roots for the original equation and which do not lose any of the roots of the original equation are said to be equivalent transformations.

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