1. simplify and write the answer in the simplest form.\na. $\frac{k}{3k - 1}+\frac{2}{3k - 1}$ b…

Explanation:

Step1: Use fraction addition - subtraction rule

When fractions have the same denominator, we add or subtract the numerators and keep the denominator. For $\frac{a}{c}\pm\frac{b}{c}=\frac{a\pm b}{c}$.

Step2: Simplify each part

a.

$\frac{k}{3k - 1}+\frac{2}{3k - 1}=\frac{k + 2}{3k - 1}$

b.

$\frac{2h}{5h - 2}-\frac{h}{5h - 2}=\frac{2h - h}{5h - 2}=\frac{h}{5h - 2}$

c.

$\frac{3t}{3t - 1}-\frac{1}{3t - 1}=\frac{3t-1}{3t - 1}=1$

d.

$\frac{2k + 1}{5k + 1}-\frac{k - 2}{5k + 1}=\frac{(2k + 1)-(k - 2)}{5k + 1}=\frac{2k+1 - k + 2}{5k + 1}=\frac{k + 3}{5k + 1}$

e.

$\frac{2y}{3y + 2}-\frac{y}{3y + 2}+\frac{1}{3y + 2}=\frac{2y-y + 1}{3y + 2}=\frac{y + 1}{3y + 2}$

f.

$\frac{2a+1}{5a - 2}-\frac{3a}{5a - 2}-\frac{3}{5a - 2}=\frac{2a + 1-3a-3}{5a - 2}=\frac{-a - 2}{5a - 2}$

g.

$\frac{8m + 10}{2m+3}-\frac{4m + 1}{2m + 3}+\frac{2m}{2m + 3}=\frac{8m + 10-(4m + 1)+2m}{2m + 3}=\frac{8m+10 - 4m - 1+2m}{2m + 3}=\frac{6m + 9}{2m + 3}=\frac{3(2m + 3)}{2m + 3}=3$

h.

$\frac{m}{m + n}-\frac{m - n}{m + n}-\frac{m - n}{m + n}=\frac{m-(m - n)-(m - n)}{m + n}=\frac{m - m + n - m + n}{m + n}=\frac{-m + 2n}{m + n}$

Answer:

a. $\frac{k + 2}{3k - 1}$ b. $\frac{h}{5h - 2}$ c. $1$ d. $\frac{k + 3}{5k + 1}$ e. $\frac{y + 1}{3y + 2}$ f. $\frac{-a - 2}{5a - 2}$ g. $3$ h. $\frac{-m + 2n}{m + n}$