Question 935881
first equation is 3x + y - 9 = 0
second equation is 2x + 9y - k = 0


solve for y in both equations.


you get:


first equation becomes y = -3x + 9
second equation becomes y = (-2x+k)/9


the x-intercept occurs when the value of y is equal to 0.


set both equations equal to 0 to get:


first equation becomes -3x + 9 = 0
second equation becomes (-2x+i)/9 = 0


solve for x in the first equation to get x = 3
solve for x in the second equation to get 2x = k


when y = 0, the value of x in both equations must be the same since both equations have the same x-intercept, so the second equation becomes 2(3) = k.


solve for k in the second equation to get k = 6.


both equations will have the same x-intercept when k = 6.


when k = 6, the second equation becomes y = (-2x + 6) / 9


the equations are now:


first equation is y = -3x + 9
second equation is y = (-2x + 6)/9


the y-intercept is the value of y when x is equal to 0.


replace x with 0 in both equations to get:


first equation becomes y = 9
second equation becomes y = 2/3.


the y-intercept for the first equation is 9.
the y intercept for the second equation is y = 2/3.


the graph of both equations is shown below:


i placed 2 horizontal lines at y = 9 and y = 2/3, so you can see the y-intercepts better.


the line that crosses the y-axis at y = 2/3 is the second equation.
the line that crosses the y-axis at y = 9 is the first equation.


the equation of the lowest horizontal line is y = 2/3.
the equation of the highest horizontal line is y = 9.


{{{graph(600,600,-10,10,-10,10,-3x+9,(-2x+6)/9,9,2/3)}}}