Solve 76=a(1.5-h)^3+k

$$76 = a ( 1.5 - h ) ^ { 3 } + k$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{k-76}{\left(1.5-h\right)^{3}}\text{, }&h\neq \frac{3}{2}\\a\in \mathrm{R}\text{, }&k=76\text{ and }h=\frac{3}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1.5-h\right)^{3}.

$$76=a\left(3.375-6.75h+4.5h^{2}-h^{3}\right)+k$$

Use the distributive property to multiply a by 3.375-6.75h+4.5h^{2}-h^{3}.

$$76=3.375a-6.75ah+4.5ah^{2}-ah^{3}+k$$

Swap sides so that all variable terms are on the left hand side.

$$3.375a-6.75ah+4.5ah^{2}-ah^{3}+k=76$$

Subtract k from both sides.

$$3.375a-6.75ah+4.5ah^{2}-ah^{3}=76-k$$

Combine all terms containing a.

$$\left(3.375-6.75h+4.5h^{2}-h^{3}\right)a=76-k$$

The equation is in standard form.

$$\left(-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375\right)a=76-k$$

Divide both sides by 3.375-6.75h+4.5h^{2}-h^{3}.

$$\frac{\left(-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375\right)a}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}=\frac{76-k}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}$$

Dividing by 3.375-6.75h+4.5h^{2}-h^{3} undoes the multiplication by 3.375-6.75h+4.5h^{2}-h^{3}.

$$a=\frac{76-k}{-h^{3}+\frac{9h^{2}}{2}-\frac{27h}{4}+3.375}$$

Divide 76-k by 3.375-6.75h+4.5h^{2}-h^{3}.

$$a=\frac{8\left(76-k\right)}{\left(3-2h\right)^{3}}$$

Solve for h

$$\left\{\begin{matrix}h=\frac{\sqrt[3]{k-76}+\frac{3\sqrt[3]{a}}{2}}{\sqrt[3]{a}}\text{, }&a\neq 0\\h\in \mathrm{R}\text{, }&k=76\text{ and }a=0\end{matrix}\right.$$