Answer:
The value of k is -4 and the equation of the line is [tex]y =-\frac{1}{2}\cdot x +10[/tex].
Step-by-step explanation:
The equation of the line is defined by the following formula:
[tex]y = m\cdot x + b[/tex] (1)
Where:
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Dependent variable.
[tex]b[/tex] - y-Intercept.
[tex]m[/tex] - Slope.
In addition, the slope of the equation of the line ([tex]m[/tex]) can be known from two distinct points:
[tex]m = \frac{y-y_{o}}{x-x_{o}}[/tex] (2)
Where [tex]x_{o}[/tex] and [tex]y_{o}[/tex] are the coordinates of the reference point.
By applying (2) in (1), we derive the resulting expression:
[tex]y = \frac{y-y_{o}}{x-x_{o}}\cdot x + b[/tex]
[tex]y\cdot (x-x_{o}) = (y-y_{o})\cdot x + b\cdot (x-x_{o})[/tex]
[tex]x\cdot y -y\cdot x_{o} = x\cdot y -x\cdot y_{o}+b\cdot x -b\cdot x_{o}[/tex]
[tex](b-y)\cdot x_{o} = b\cdot x -x\cdot y_{o}[/tex]
[tex](b-y)\cdot x_{o} +x\cdot y_{o} = b\cdot x[/tex] (3)
If we know that [tex]b = 10[/tex], [tex]x = 2[/tex], [tex]y = 9[/tex], [tex]x_{o} = k+10[/tex] and [tex]y_{o} = -2\cdot k -1[/tex], then the value of [tex]k[/tex] is:
[tex](10-9)\cdot (k+10)+2\cdot (-2\cdot k -1) = (10)\cdot (2)[/tex]
[tex]k+10 -4\cdot k -2 = 20[/tex]
[tex]-3\cdot k +8 = 20[/tex]
[tex]-3\cdot k = 12[/tex]
[tex]k = -4[/tex]
The value of the slope is:
[tex]m = \frac{9-7}{2-6}[/tex]
[tex]m = -\frac{1}{2}[/tex]
The equation of the line is [tex]y =-\frac{1}{2}\cdot x +10[/tex].
