Given the equation [tex]\(-cz + 6z = tz + 83\)[/tex], let's solve for [tex]\(z\)[/tex].
1. Combine like terms: On the left side of the equation, combine the terms involving [tex]\(z\)[/tex].
[tex]\[
-cz + 6z
\][/tex]
The right side of the equation remains as:
[tex]\[
tz + 83
\][/tex]
2. Let's rewrite the equation:
[tex]\[
(-c + 6)z = tz + 83
\][/tex]
3. Move all terms involving [tex]\(z\)[/tex] to one side: Here, we subtract [tex]\(tz\)[/tex] from both sides of the equation to combine all the [tex]\(z\)[/tex]-terms on one side.
[tex]\[
(-c + 6)z - tz = 83
\][/tex]
4. Factor out [tex]\(z\)[/tex] from the left side of the equation:
[tex]\[
z(-c + 6 - t) = 83
\][/tex]
5. Simplify the expression in the parenthesis:
[tex]\[
z(-c - t + 6) = 83
\][/tex]
6. Solve for [tex]\(z\)[/tex]: To isolate [tex]\(z\)[/tex], divide both sides of the equation by the expression [tex]\((-c - t + 6)\)[/tex]:
[tex]\[
z = \frac{83}{-c - t + 6}
\][/tex]
Thus, the solution for [tex]\(z\)[/tex] in the given equation [tex]\(-cz + 6z = tz + 83\)[/tex] is:
[tex]\[
z = \frac{83}{-c - t + 6}
\][/tex]
This is a step-by-step solution showing how to isolate and solve for the variable [tex]\(z\)[/tex] in the given equation.
