Writing Slope-Intercept Equations

1.\(y=-3 x+7\)

Let's find the slope:

\(\begin{aligned}

\text { Slope } &=\frac{(-8)-1}{5-2} \\

&=\frac{-9}{3} \\

&=-3

\end{aligned}\)

The equation is \(y=-3 x+b\) for some \(b\).

Let's plug the point \( (2,1)\) to find \(b\):

\(\begin{aligned}

&y=-3 x+b \\

&1=-3(2)+b \\

&1=-6+b \\

&7=b

\end{aligned}\)

The equation is \(y=-3 x+7\).

2. \(y=\frac{4}{3} x-12\)

Let's find the slope:

\(\begin{aligned}

\text { Slope } &=\frac{-4-(-8)}{6-3} \\

&=\frac{4}{3}

\end{aligned}\)

The equation is \(y=\frac{4}{3} x+b \) for some \(b\).

Let's plug the point \((6,−4) \)to find \(b\):

\(\begin{aligned}

y &=\frac{4}{3} x+b \\

-4 &=\frac{4}{3}(6)+b \\

-4 &=8+b \\

-12 &=b

\end{aligned}\)

The equation is \(y=\frac{4}{3} x-12\).

3. \(y=8 x-25\).

Let's find the slope:

\(\begin{aligned}

\text { Slope } &=\frac{7-(-1)}{4-3} \\

&=\frac{8}{1} \\

&=8

\end{aligned}\)

The equation is \(y=8 x+b \) for some \(b\).

Let's plug the point \((4, 7)\) to find \(b\):

\(\begin{aligned}

y &=8 x+b \\

7 &=8(4)+b \\

7 &=32+b \\

-25 &=b

\end{aligned}\)

The equation is \(y=8 x-25\).

4. \(y=-\frac{2}{5} x-11\)

Let's find the slope:

\(\begin{aligned}

\text { Slope } &=\frac{-9-(-7)}{-5-(-10)} \\

&=\frac{-2}{5} \\

&=-\frac{2}{5}

\end{aligned}\)

The equation is \(y=-\frac{2}{5} x+b\) for some \(b\).

Let's plug the point \( (−5,−9) \) to find \(b\):

\(\begin{aligned}

y &=-\frac{2}{5} x+b \\

-9 &=-\frac{2}{5}(-5)+b \\

-9 &=2+b \\

-11 &=b

\end{aligned}\)

The equation is \(y=-\frac{2}{5} x-11\).