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Mathematics

Verify that x×(y−z)=x×y−x×zx \times (y - z) = x \times y - x \times z, if

x=45,y=−74x = \dfrac{4}{5}, y = \dfrac{-7}{4} and z=3z = 3

Rational Numbers

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Answer

To prove:

x×(y−z)=x×y−x×zx \times (y - z) = x \times y - x \times z

Taking LHS:

x×(y−z)=45×(−74−3)=45×(−74−31)x \times (y - z)\\[1em] =\dfrac{4}{5} \times \Big(\dfrac{-7}{4} - 3\Big)\\[1em] =\dfrac{4}{5} \times \Big(\dfrac{-7}{4} - \dfrac{3}{1}\Big)

LCM of 4 and 1 is 2 x 2 = 4.

=45×(−7×14×1−3×41×4)=45×(−74−124)=45×(−7−124)=45×(−194)=(4×−195×4)=(−7620)=(−195)=\dfrac{4}{5} \times \Big(\dfrac{-7 \times 1}{4 \times 1} - \dfrac{3 \times 4}{1 \times 4}\Big)\\[1em] =\dfrac{4}{5} \times \Big(\dfrac{-7}{4} - \dfrac{12}{4}\Big)\\[1em] =\dfrac{4}{5} \times \Big(\dfrac{-7 - 12}{4}\Big)\\[1em] =\dfrac{4}{5} \times \Big(\dfrac{-19}{4}\Big)\\[1em] =\Big(\dfrac{4 \times -19}{5 \times 4}\Big)\\[1em] =\Big(\dfrac{-76}{20}\Big)\\[1em] =\Big(\dfrac{-19}{5}\Big)

Taking RHS:

x×y−x×z=45×−74−45×3=4×−75×4−4×35×1=−2820−125=−75−125=−7−125=−195x \times y - x \times z\\[1em] =\dfrac{4}{5} \times \dfrac{-7}{4} - \dfrac{4}{5} \times 3\\[1em] = \dfrac{4 \times -7}{5 \times 4} - \dfrac{4 \times 3}{5 \times 1}\\[1em] =\dfrac{-28}{20} - \dfrac{12}{5}\\[1em] =\dfrac{-7}{5} - \dfrac{12}{5}\\[1em] =\dfrac{-7 - 12}{5}\\[1em] =\dfrac{-19}{5}

∴ LHS = RHS

x×(y−z)=x×y−x×zx \times (y - z) = x \times y - x \times z

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