[tex]\text{Solve for z:}\\\\-p(51+z)=dz+84\\\\\text{Use the distributive property}\\\\-51p-pz=dz+84\\\\\text{Add 51p to both sides}\\\\-pz=51p+dz+84\\\\\text{Subtract dz from both sides}\\\\-pz-dz=51p+84\\\\\text{Factor out z}\\\\z(-d-p)=51p+84\\\\\text{Divide both sides by (-d - p)}\\\\z=\frac{51p+84}{(-d-p)}\\\\\text{The denominator shouldn't be zero, so we have to make it positive}\\\\\boxed{z=\frac{-51p-84}{d+p}\,\,or\,\,z=-\frac{51p+84}{d+p}}[/tex]
Answer:
[tex] \boxed{\sf z = \frac{ - 51p - 84}{ p + d} \ \ \ OR \ \ \ z = - \frac{51p + 84}{ p + d} } [/tex]
Step-by-step explanation:
[tex] \sf solve \: for \: z : \\ \sf \implies −p(51+z)=dz+84 \\ \\ \sf Expand \: out \: terms \: of \: the \: left \: hand \: side: \\ \sf \implies - 51p - pz = dz + 84 \\ \\ \sf Subtract \: d z - 51 p \: from \: both \: sides: \\ \sf \implies - 51p - pz - (dz - 51p)= dz + 84 - (dz - 51p) \\ \\ \sf - (dz - 51p) = - dz + 51p : \\ \sf \implies - 51p - pz - dz + 51p= dz + 84 - dz + 51p \\ \\ \sf - 51p + 51p = 0 : \\ \sf \implies - pz - dz = dz + 84 - dz + 51p \\ \\ \sf dz - dz = 0 : \\ \sf \implies - pz - dz = 84 + 51p \\ \\ \sf \implies z( - p - d) = 84 + 51p \\ \\ \sf Divide \: both \: sides \: by \: - p - d: \\ \sf \implies z = \frac{51p + 84}{ - p - d} \\ \\ \sf \implies z = \frac{51p + 84}{ -( p + d)} \\ \\ \sf \implies z = \frac{ - (51p + 84)}{ p + d} \\ \\ \sf \implies z = \frac{ - 51p - 84}{ p + d} [/tex]
