Question

PLZ HELP ;(((

A line passes through the points (k + 10,-2k – 1) and (2,9) and has a y-intercept
of 10. Find the value of k and the equation of the line.
solve for all 3 equations

Answer 1

The value of k is -4 and the equation of the line is
`y =-(1)/(2)\cdot x +10`.

Explanation:

The equation of the line is defined by the following formula:

`y = m\cdot x + b` (1)

Where:

`x` - Independent variable.

`y` - Dependent variable.

`b` - y-Intercept.

`m` - Slope.

In addition, the slope of the equation of the line (
`m`) can be known from two distinct points:

`m = (y-y_(o))/(x-x_(o))` (2)

Where
`x_(o)` and
`y_(o)` are the coordinates of the reference point.

By applying (2) in (1), we derive the resulting expression:

`y = (y-y_(o))/(x-x_(o))\cdot x + b`

`y\cdot (x-x_(o)) = (y-y_(o))\cdot x + b\cdot (x-x_(o))`

`x\cdot y -y\cdot x_(o) = x\cdot y -x\cdot y_(o)+b\cdot x -b\cdot x_(o)`

`(b-y)\cdot x_(o) = b\cdot x -x\cdot y_(o)`

`(b-y)\cdot x_(o) +x\cdot y_(o) = b\cdot x` (3)

If we know that
`b = 10`,
`x = 2`,
`y = 9`,
`x_(o) = k+10` and
`y_(o) = -2\cdot k -1`, then the value of
`k` is:

`(10-9)\cdot (k+10)+2\cdot (-2\cdot k -1) = (10)\cdot (2)`

`k+10 -4\cdot k -2 = 20`

`-3\cdot k +8 = 20`

`-3\cdot k = 12`

`k = -4`

The value of the slope is:

`m = (9-7)/(2-6)`

`m = -(1)/(2)`

The equation of the line is
`y =-(1)/(2)\cdot x +10`.