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Generalization to fractions is by multiplying the numerators and denominators, respectively: = (). This gives the area of a rectangle A B {\displaystyle {\frac {A}{B}}} high and C D {\displaystyle {\frac {C}{D}}} wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.
While not normally taught as a standard method for multiplying fractions, the grid method can readily be applied to simple cases where it is easier to find a product by breaking it down. For example, the calculation 2 1 / 2 × 1 1 / 2 can be set out using the grid method
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
The solution set for the equations = and + = is the single point (2, 3). An example of solving a system of linear equations is by using the elimination method: {+ = = Multiplying the terms in the second equation by 2:
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