First, multiply each side of the equation by the common denominator of the two fractions which is color(red)(5)color(blue)(z). This will eliminate the fractions while keeping the equation balanced:
color(red)(5)color(blue)(z) xx 6/z = color(red)(5)color(blue)(z) xx 3/5
color(red)(5)cancel(color(blue)(z)) xx 6/color(blue)(cancel(color(black)(z))) = cancel(color(red)(5))color(blue)(z) xx 3/color(red)(cancel(color(black)(5)))
30 = 3z
Now, divide each side of the equation by color(red)(3) to solve for z while keeping the equation balanced:
30/color(red)(3) = (3z)/color(red)(3)
10 = (color(red)(cancel(color(red)(3)))z)/cancel(color(red)(3))
10 = z
z = 10
Another method to solve an equation of two fractions is to "flip" the fractions and then solve for z:
z/6 = 5/3
Now multiply each side of the equation by color(red)(6) to solve for z:
color(red)(6) xx z/6 = color(red)(6) xx 5/3
cancel(color(red)(6)) xx z/color(red)(cancel(color(black)(6))) = 30/3
z = 10