(i)LCM of 5 and 7 = 35
\(\frac{-12}{5}= \frac{-12\times7}{5\times7} = \frac{-84}{35}\)
And,
\(\frac{2}{7}= \frac{2\times5}{7\times5} = \frac{10}{35}\)
LHS = \(\frac{-12}{5}+\frac{2}{7} = \frac{-84}{35}+\frac{10}{35}=\) \(\frac{-84+10}{35} = \frac{-74}{35}\)
Similarly,
LCM of 7 and 5 = 35
\(\frac{2}{7}= \frac{2\times5}{7\times5} = \frac{10}{35}\)
And,
\(\frac{-12}{5}= \frac{-12\times7}{5\times7} = \frac{-84}{35}\)
RHS = \(\frac{2}{7}+ \frac{-12}{5}= \frac{10}{35}+\frac{-84}{35}\) = \(\frac{10+(-84)}{35} = \frac{10-84}{35}= \frac{-74}{35}\)
i.e., LHS = RHS
Hence,
\(\frac{-12}{5}+\frac{2}{7}= \frac{2}{7}+\frac{-12}{5}\)
Verified
\(\frac{-9}{13}+\frac{-9\times8}{13\times8 }= \frac{-72}{104}\)
LHS = \(\frac{-5}{8}+ \frac{-9}{13}= \frac{-65}{104}+\frac{-72}{104}\) = \(\frac{-65+(-84)}{104} = \frac{-65-72}{104}= \frac{-137}{104}\)
Similarly,
(ii)LCM of 8 and 13 = 104
\(\frac{-5}{8}= \frac{-5\times13}{8\times13}= \frac{-65}{104}\)
And,
\(\frac{-9}{13}+\frac{-9\times8}{13\times8 }= \frac{-72}{104}\)
RHS = \(\frac{-9}{13}+ \frac{-5}{8}= \frac{-72}{104}+\frac{-65}{104}\) = \(\frac{-72+(-65)}{35} = \frac{-72-65}{35}= \frac{-137}{104}\)
i.e., LHS = RHS
Hence,
\(\frac{-5}{8}+\frac{-9}{13}= \frac{-9}{13}+\frac{-5}{8}\)
Verified
(iii) 3 can be written as \(\frac{3}{1}\)
LCM of 1 and 12 = 12
\(\frac{2}{1}= \frac{3\times12}{1\times12} = \frac{36}{12}\)
And,
\(\frac{2}{1}= \frac{3\times12}{1\times12} = \frac{36}{12}\)
\(\frac{-7}{12}= \frac{-7\times1}{12\times1}= \frac{-7}{12}\)
LHS = \(\frac{3}{1}+ \frac{-7}{12}= \frac{36}{12}+\frac{-7}{12}\) = \(\frac{36+(-7)}{12} = \frac{36-7}{12}= \frac{29}{12}\)
Similarly,
LCM of 1 and 12 = 12
\(\frac{-7}{12}= \frac{-7\times1}{12\times1}= \frac{-7}{12}\)
And,
\(\frac{3}{1}= \frac{3\times12}{1\times12} = \frac{36}{12}\)
RHS = \(\frac{-7}{12}+\frac{3}{1}= \frac{-7}{12}+\frac{36}{12}= \) \(\frac{-7+36}{12} = \frac{29}{12}\)
i.e., LHS = RHS
Hence,
\(3+\frac{-7}{12} = \frac{-7}{12}+3\)
Verified
(iv) Since, the denominators are negative we will make them positive.
\(\frac{2}{-7} =\frac{2\times-1}{-7\times-1} = \frac{-2}{7}\)
And,
\(\frac{12}{-35} = \frac{12\times-1}{35\times-1} = \frac{-12}{35}\)
LCM of 7 and 35 = 35
\(\frac{-2}{7} =\frac{-2\times5}{7\times5} = \frac{-10}{35}\)
And,
\(\frac{-12}{35} =\frac{-12\times1}{35\times1} = \frac{-12}{35}\)
LHS = \(\frac{-2}{7}+\frac{-12}{35}= \frac{-10}{35}+\frac{-12}{35}= \) \(\frac{-10+(-12)}{35} = \frac{-10-12}{35} = \frac{-22}{35}\)
Similarly,
LCM of 7 and 5 = 35
\(\frac{-12}{35} =\frac{-12\times1}{35\times1} = \frac{-12}{35}\)
RHS = \(\frac{-12}{35}+\frac{-2}{7}= \frac{-12}{35}+\frac{-10}{35}\) = \(\frac{-12+(-10)}{35}\) = \(\frac{-12-10}{35}= \frac{-22}{35}\)
i.e., LHS = RHS
Hence,
\(\frac{-2}{7}+\frac{-12}{35}=\frac{-12}{35}+\frac{-2}{7}\)
