You have the following equation:
5x + 3y = 30
solve the previous equation for y and for x, just as follow:
5x + 3y = 30
5x = -3y + 30
x = -3/5 y + 6
5x + 3y = 30
3y = -5x + 30
y = -5/3 x + 10
the x-intercept is the value of x when y=0:
x = -3/5 (0) + 6
x = 6 x-intercept
the y-intercept is the value of y when x=0:
y = -5/3(0) + 10
y = 10 y-intercept