To solve, let's consider the given slope of a line.
`m=-1`
Take note that when a line is tangent to the curve, its derivative at the point of tangency is equal to the slope.
`y' = m`
`y'=1`
To determine the point of tangency, take the derivative of the function of the curve.
`y= 1/(x-1)`
`y=(x-1)^(-1)`
`y'=-1(x-1)^(-1-1) * (x-1)'`
`y'=-(x-1)^(-2)*(1)`
`y'=-1/(x-1)^2`
Then, plug-in the value of the derivative.
`-1=-1/(x-1)^2`
`1=1/(x-1)^2`
`(x-1)^2=1`
`x-1=+-1`
`x=+-1+1`
`x_1=0`
`x_2=2`
To solve for the values of y, plug-in x=0 and x=2.
`x_1=0`
`y_1=1/(0-1)=-1`
`x_2=2`
`y_2=1/(2-1)=1`
So, there two points on the curve in which the tangent have a slope of -1. These are:
(0,-1) and (2,1).
This means that there are two lines that have a slope of -1. To determine the equation of each line, use the point-slope form.
The first line has a slope of -1 and passes the point (0,-1). So its equation is:
`y - y_1 = m(x - x_1)`
`y - (-1) = -1(x-0)`
`y+1=-x`
`y=-x - 1`
The second line has a slope of -1 and passes the point (2,1). So its equation is:
`y - y_2=m(x- x_2)`
`y-1 = -1(x-2)`
`y-1=-x+2`
`y=-x+3`
Therefore, the equation of the lines with slope -1 and tangent to the given curve are:
`y=-x-1`
and
`y=-x+3` .
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