Question

Verify the following:
(i) $\frac{-12}{5}+\frac{2}{7}=\frac{2}{7}+\frac{-12}{5}$
(ii) $\frac{-5}{8}+\frac{-9}{13}=\frac{-9}{13}+\frac{-5}{8}$
(iii) $3+\frac{-7}{12}=\frac{-7}{12}+3$
(iv) $\frac{2}{-7}+\frac{12}{-35}=\frac{12}{-35}+\frac{2}{-7}$

Answer 1

1. LHS = $\frac{-12}{5}+\frac{2}{7}$

LCM of 5 and 7 is 35.

$\frac{-12\times 7}{5\times 7}+\frac{2\times 5}{7\times 5}=\frac{-84}{35}+\frac{10}{35}=\frac{-84+10}{35}=\frac{-74}{35}$

RHS = $\frac{2}{7}+\frac{-12}{5}$

LCM of 5 and 7 is 35.

$\frac{2\times 5}{7\times 5}+\frac{-12\times 7}{5\times 7}=\frac{10}{35}+\frac{-84}{35}=\frac{10-84}{35}=\frac{-74}{35}$

∴ $\frac{-12}{5}+\frac{2}{7}=\frac{2}{7}+\frac{-12}{5}$

2. ​LHS = $\frac{-5}{8}+\frac{-9}{13}$

LCM of 8 and 13 is 104.

$\frac{-5\times 13}{8\times 13}+\frac{-9\times 8}{13\times 8}=\frac{-65}{104}+\frac{-72}{104}=\frac{-65+(-72)}{104}=\frac{-65-72}{104}=\frac{-137}{104}$

RHS = $\frac{-9}{13}+\frac{-5}{8}$

LCM of 13 and 8 is 104.

$\frac{-9\times 8}{13\times 8}+\frac{-5\times 13}{8\times 13}=\frac{-72}{104}+\frac{-65}{104}=\frac{-72-65}{104}=\frac{-137}{104}$

∴ $\frac{-5}{8}+\frac{-9}{13}=\frac{-9}{13}+\frac{-5}{8}$

3. ​LHS = $\frac{3}{1}+\frac{-7}{12}$

LCM of 1 and 12 is 12.

$\frac{3\times 12}{1\times 12}+\frac{-7\times 1}{12\times 1}=\frac{36}{12}+\frac{-7}{12}=\frac{36+(-7)}{12}=\frac{36-7}{12}=\frac{29}{12}$

RHS = $\frac{-7}{12}+\frac{3}{1}$

LCM of 12 and 1 is 12.

$\frac{-7\times 1}{12\times 1}+\frac{3\times 12}{1\times 12}=\frac{-7}{12}+\frac{36}{12}=\frac{-7+36}{12}=\frac{29}{12}$

∴ $3+\frac{-7}{12}=\frac{-7}{12}+3$

4. LHS = ​$\frac{2}{-7}+\frac{12}{-35}$

We will write the given numbers with positive denominators.

$\frac{2}{-7}=\frac{2\times (-1)}{-7\times (-1)}=\frac{-2}{7}$ and $\frac{12}{-35}=\frac{12\times (-1)}{-35\times (-1)}=\frac{-12}{35}$

LCM of 7 and 35 is 35.

$\frac{-2\times 5}{7\times 5}+\frac{-12\times 1}{35\times 1}=\frac{-10}{35}+\frac{-12}{35}=\frac{-10+(-12)}{35}=\frac{-10-12}{35}=\frac{-22}{35}$

RHS = $\frac{12}{-35}+\frac{2}{-7}$

We will write the given numbers with positive denominators.

$\frac{12}{-35}=\frac{12\times (-1)}{-35\times (-1)}=\frac{-12}{35}$ and $\frac{2}{-7}=\frac{2\times (-1)}{-7\times (-1)}=\frac{-2}{7}$

LCM of 35 and 7 is 35.

$\frac{-2\times 5}{7\times 5}+\frac{-12\times 1}{35\times 1}=\frac{-10}{35}+\frac{-12}{35}=\frac{-10+(-12)}{35}=\frac{-10-12}{35}=\frac{-22}{35}$

∴​ $\frac{2}{-7}+\frac{12}{-35}=\frac{12}{-35}+\frac{2}{-7}$