First, multiply each side of the equation by color(red)(30)30 to eliminate the fractions while keeping the equation balanced. color(red)(3)3 is the Lowest Common Denominator of the three fractions:
color(red)(30)(1/2z + 1/3) = color(red)(30) * -2/530(12z+13)=30⋅−25
(color(red)(30) * 1/2z) + (color(red)(30) * 1/3) = cancel(color(red)(30))6 * -2/color(red)(cancel(color(black)(5)))
(cancel(color(red)(30)) 15 * 1/color(red)(cancel(color(black)(2)))z) + (cancel(color(red)(30)) 10 * 1/color(red)(cancel(color(black)(3)))) = 6 * -2
(15 * 1z) + (10 * 1) = -12
15z + 10 = -12
Next, subtract color(red)(10) from each side of the equation to isolate the z term while keeping the equation balanced:
15z + 10 - color(red)(10) = -12 - color(red)(10)
15z + 0 = -22
15z = -22
Now, divide each side of the equation by color(red)(15) to solve for z while keeping the equation balanced:
(15z)/color(red)(15) = -22/color(red)(15)
(color(red)(cancel(color(black)(15)))z)/cancel(color(red)(15)) = -22/15
z = -22/15