Step 1

Equation of line that passes through the points \(\displaystyle{\left({x}_{{{1}}},\ {y}_{{{1}}}\right)}\ \ {\left({x}_{{{2}}},\ {y}_{{{2}}}\right)}\) is:

\(\displaystyle{y}-{y}_{{{1}}}={\frac{{{\left({y}_{{{2}}}-{y}_{{{1}}}\right)}}}{{{\left({x}_{{{2}}}-{x}_{{{1}}}\right)}}}}{\left({\left({x}-{x}_{{{1}}}\right)}\right.}\)

where,

slope \(\displaystyle{m}={\frac{{{\left({y}_{{{2}}}-{y}_{{{1}}}\right)}}}{{{\left({x}_{{{2}}}-{x}_{{{1}}}\right)}}}}\)

Step 2

a) Given points \(\displaystyle{\left({2},\ {4}\right)}\ \ {\left({4},\ {7}\right)}\)

now

\(\displaystyle{x}_{{{1}}}={2}\)

\(\displaystyle{y}_{{{1}}}={4}\)

\(\displaystyle{x}_{{{2}}}={4}\)

\(\displaystyle{y}_{{{2}}}={7}\)

Equation of a line:

\(\displaystyle{y}-{4}={\frac{{{\left({7}-{4}\right)}}}{{{\left({4}-{2}\right)}}}}{\left({\left({x}-{2}\right)}\right.}\)

\(\displaystyle{y}-{4}={\frac{{{3}}}{{{2}}}}{\left({x}-{2}\right)}\)

\(\displaystyle{y}={\frac{{{3}}}{{{2}}}}{x}-{\frac{{{6}}}{{{2}}}}+{4}\)

\(\displaystyle{y}={\frac{{{3}}}{{{2}}}}{x}+{1}\)

Step 3

Given points \(\displaystyle{\left({3},\ -{5}\right)}\ \ {\left(-{2},\ {2}\right)}\)

now

\(\displaystyle{x}_{{{1}}}={3}\)

\(\displaystyle{y}_{{{1}}}=-{5}\)

\(\displaystyle{x}_{{{2}}}=-{2}\)

\(\displaystyle{y}_{{{2}}}={2}\)

Equation of a line:

\(\displaystyle{y}+{5}={\frac{{{\left({2}+{5}\right)}}}{{{\left(-{2}-{3}\right)}}}}{\left({x}-{3}\right)}\)

\(\displaystyle{y}+{5}=-{\frac{{{7}}}{{{5}}}}{\left({x}-{3}\right)}\)

\(\displaystyle{y}=-{\frac{{{7}}}{{{5}}}}{x}+{\frac{{{21}}}{{{5}}}}-{5}\)

\(\displaystyle{y}=-{\frac{{{7}}}{{{5}}}}{x}-{\frac{{{4}}}{{{5}}}}\)

Step 4

Given points \(\displaystyle{\left({1},\ {3}\right)}\ {\left(-{3},\ {1}\right)}\)

now

\(\displaystyle{x}_{{{1}}}={1}\)

\(\displaystyle{y}_{{{1}}}={3}\)

\(\displaystyle{x}_{{{2}}}=-{3}\)

\(\displaystyle{y}_{{{2}}}={1}\)

Equation of a line:

\(\displaystyle{y}-{3}={\frac{{{\left({1}-{3}\right)}}}{{{\left(-{3}-{1}\right)}}}}{\left({x}-{1}\right)}\)

\(\displaystyle{y}-{3}={\frac{{{1}}}{{{2}}}}{\left({x}-{1}\right)}\)

\(\displaystyle{y}={\frac{{{1}}}{{{2}}}}{x}-{\frac{{{1}}}{{{2}}}}+{3}\)

\(\displaystyle{y}={\frac{{{1}}}{{{2}}}}{x}+{\frac{{{5}}}{{{2}}}}\)