Question 888907
you want to to convert both equation into slope intercept form.


slope intercept form is y = mx + b


once in this form:


m = slope
b = y-intercept.


if the slope of both equation is the same and the y-intercept is different, then the lines are parallel.


start with x + 5y + 2 = 0
subtract x and subtract 2 from both sides of the equation to get:
5y = -x - 2
divide both sides of the equation by 5 to get:
y = -(1/5)*x - (2/5)


that's your first equation in slope intercept form.
the slope is -(1/5).


-(1/5) is the slope you want to get in your second equation in order to make the lines parallel to each other.


start with your second equation of x + (3k-2)y + 10 = 0
subtract 10 and subtract x from both sides of the equation to get:
(3k-2)y = -x - 10
divide both sides of the equation by (3k-2) to get:
y = -(1/(3k-2)*x - (10/(3k-2))


that's your second equation in slope intercept form.
the slope is -(1/(3k-2))


in order for the lines to be parallel, -(1/(3k-2)) must be equal to -(1/5).


set these two expressions equal to each other and solve for k.


-1/(3k-2) = -1/5


cross multiply to get:


-1*5 = -1*(3k-2)


divide both sides of this equation by (-1) to get:


5 = 3k-2


add 2 to both sides of this equation to get:


7 = 3k


divide both sides of this equation by 3 to get:


7/3 = k


that's your answer.


in order for the lines to be parallel, k has to be equal to 7/3.


to confirm, replace k with 7/3 in your original equation.


your original equation is x + (3k-2)y + 10 = 0
replace k with 7/3 to get:
x + (3*7/3-2)y + 10 = 0
simplify to get:
x + 5y + 10 = 0
subtract x and subtract 10 from both sides of the equation to get:
5y = -x - 10
divide both sides of the equation by 5 to get:
y = -(1/5)*x - (10/5)
simplify to get:
y = -(1/5)*x - 2


your first equation in slope intercept form is y = -(1/5)*x - (2/5)


your second equation in slope intercept form is y = -(1/5)*x - 2


the slopes are the same and the y-intercepts are different so the lines are parallel.


here's a graph of your two lines.
{{{graph(500,500,-10,10,-10,10,-(1/5)*x - (2/5),-(1/5)*x - 2)}}}