Solve for z -p*(51+z)=dz+84

$- p \cdot (51 + z) = d z + 84$

Simplify $- p \cdot (51 + z)$ .

Rewrite.

$0 + 0 - p \cdot (51 + z) = d z + 84$

Simplify by adding zeros.

$- p \cdot (51 + z) = d z + 84$

Apply the distributive property.

$- p \cdot 51 - p z = d z + 84$

Multiply $51$ by $- 1$ .

$- 51 p - p z = d z + 84$

Subtract $d z$ from both sides of the equation.

$- 51 p - p z - d z = 84$

Add $51 p$ to both sides of the equation.

$- p z - d z = 84 + 51 p$

Factor $z$ out of $- p z - d z$ .

Factor $z$ out of $- p z$ .

$z (- p) - d z = 84 + 51 p$

Factor $z$ out of $- d z$ .

$z (- p) + z (- d) = 84 + 51 p$

Factor $z$ out of $z (- p) + z (- d)$ .

$z (- p - d) = 84 + 51 p$

Divide each term in $z (- p - d) = 84 + 51 p$ by $- p - d$ and simplify.

Divide each term in $z (- p - d) = 84 + 51 p$ by $- p - d$ .

$\frac{z (- p - d)}{- p - d} = \frac{84}{- p - d} + \frac{51 p}{- p - d}$

Simplify the left side.

Cancel the common factor of $- p - d$ .

Cancel the common factor.

$\frac{z (- p - d)}{- p - d} = \frac{84}{- p - d} + \frac{51 p}{- p - d}$

Divide $z$ by $1$ .

$z = \frac{84}{- p - d} + \frac{51 p}{- p - d}$

Simplify the right side.

Combine the numerators over the common denominator.

$z = \frac{84 + 51 p}{- p - d}$

Factor $3$ out of $84 + 51 p$ .

Factor $3$ out of $84$ .

$z = \frac{3 (28) + 51 p}{- p - d}$

Factor $3$ out of $51 p$ .

$z = \frac{3 (28) + 3 (17 p)}{- p - d}$

Factor $3$ out of $3 (28) + 3 (17 p)$ .

$z = \frac{3 (28 + 17 p)}{- p - d}$

Factor $- 1$ out of $- p$ .

$z = \frac{3 (28 + 17 p)}{- (p) - d}$

Factor $- 1$ out of $- d$ .

$z = \frac{3 (28 + 17 p)}{- (p) - (d)}$

Factor $- 1$ out of $- (p) - (d)$ .

$z = \frac{3 (28 + 17 p)}{- (p + d)}$

Rewrite negatives.

Rewrite $- (p + d)$ as $- 1 (p + d)$ .

$z = \frac{3 (28 + 17 p)}{- 1 (p + d)}$

Move the negative in front of the fraction.

$z = - \frac{3 (28 + 17 p)}{p + d}$

Solve for z -p*(51+z)=dz+84

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