Answer
Point-slope form is $y-2=2\left( x-1 \right)$.
Slope-intercept form is $y=2x$.
Work Step by Step
The point slope form of a line can be obtained with the help of the slope of the line and any one point that lies on the line.
For a line with slope m and passing through the point $\left( {{x}_{1}},{{y}_{1}} \right)$ , the point-slope form is given by
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Consider $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 1,2 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 5,10 \right)$
At first, calculate the slope of the line by $m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ ; here, ${{x}_{2}}-{{x}_{1}}\ne 0$
Substitute the value of $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ in it:
$\begin{align}
& m=\frac{10-2}{5-1} \\
& =\frac{8}{4} \\
& =2
\end{align}$
The point-slope form is $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$.
Replace the values of m and $\left( {{x}_{1}},{{y}_{1}} \right)$ in the above equation to get
$\begin{align}
& y-2=2\left( x-1 \right) \\
& y-2=2x-2 \\
& y=2x-2+2 \\
& =2x
\end{align}$
The point-slope form of the equation is $y=2x$.
The slope-intercept form of the line is given by $y=mx+b$ ; here, m is the slope and b is the y-intercept, and the y-intercept is the y-coordinate of a point where the line intersects the y-axis.
Substitute the value of m and take the point $\left( x,y \right)=\left( 1,2 \right)$ in $y=mx+b$ to get the value of b:
$\begin{align}
& 2=2\left( 1 \right)+b \\
& b=0
\end{align}$
Now, replace the values of b and m in $y=mx+b$.
Thus
$y=2x+0$
The slope-intercept form of the line is $y=2x$.
Hence, the point-slope form of line passing through the points $\left( 1,2 \right)$ and $\left( 5,10 \right)$ is $y-2=2\left( x-1 \right)$ , and the slope-intercept form is $y=2x$.
