Solve -(m+1)(n-4)/sqrt{4^2+m^2}=frac{4sqrt{2}}{sqrt{17}}

$$\frac { - ( m + 1 ) ( n - 4 ) } { \sqrt { 4 ^ { 2 } + m ^ { 2 } } } = \frac { 4 \sqrt { 2 } } { \sqrt { 17 } }$$

Solve for n

$$n=\frac{4\left(-\sqrt{34\left(m^{2}+16\right)}+17m+17\right)}{17\left(m+1\right)}$$

$$m\neq -1$$

Steps for Solving Linear Equation

To find the opposite of m+1, find the opposite of each term.

$$\frac{\left(-m-1\right)\left(n-4\right)}{\sqrt{4^{2}+m^{2}}}=\frac{4\sqrt{2}}{\sqrt{17}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(-m-1\right)\left(n-4\right)}{\sqrt{16+m^{2}}}=\frac{4\sqrt{2}}{\sqrt{17}}$$

Rationalize the denominator of \frac{4\sqrt{2}}{\sqrt{17}} by multiplying numerator and denominator by \sqrt{17}.

$$\frac{\left(-m-1\right)\left(n-4\right)}{\sqrt{16+m^{2}}}=\frac{4\sqrt{2}\sqrt{17}}{\left(\sqrt{17}\right)^{2}}$$

The square of \sqrt{17} is 17.

$$\frac{\left(-m-1\right)\left(n-4\right)}{\sqrt{16+m^{2}}}=\frac{4\sqrt{2}\sqrt{17}}{17}$$

To multiply \sqrt{2} and \sqrt{17}, multiply the numbers under the square root.

$$\frac{\left(-m-1\right)\left(n-4\right)}{\sqrt{16+m^{2}}}=\frac{4\sqrt{34}}{17}$$

Use the distributive property to multiply -m-1 by n-4.

$$\frac{-mn+4m-n+4}{\sqrt{16+m^{2}}}=\frac{4\sqrt{34}}{17}$$

Multiply both sides of the equation by 17.

$$17\left(16+m^{2}\right)^{-\frac{1}{2}}\left(-mn+4m-n+4\right)=4\sqrt{34}$$

Reorder the terms.

$$17\left(m^{2}+16\right)^{-\frac{1}{2}}\left(-mn+4m-n+4\right)=4\sqrt{34}$$

Use the distributive property to multiply 17\left(m^{2}+16\right)^{-\frac{1}{2}} by -mn+4m-n+4.

$$-17n\left(m^{2}+16\right)^{-\frac{1}{2}}m+68\left(m^{2}+16\right)^{-\frac{1}{2}}m-17n\left(m^{2}+16\right)^{-\frac{1}{2}}+68\left(m^{2}+16\right)^{-\frac{1}{2}}=4\sqrt{34}$$

Subtract 68\left(m^{2}+16\right)^{-\frac{1}{2}}m from both sides.

$$-17n\left(m^{2}+16\right)^{-\frac{1}{2}}m-17n\left(m^{2}+16\right)^{-\frac{1}{2}}+68\left(m^{2}+16\right)^{-\frac{1}{2}}=4\sqrt{34}-68\left(m^{2}+16\right)^{-\frac{1}{2}}m$$

Subtract 68\left(m^{2}+16\right)^{-\frac{1}{2}} from both sides.

$$-17n\left(m^{2}+16\right)^{-\frac{1}{2}}m-17n\left(m^{2}+16\right)^{-\frac{1}{2}}=4\sqrt{34}-68\left(m^{2}+16\right)^{-\frac{1}{2}}m-68\left(m^{2}+16\right)^{-\frac{1}{2}}$$

Combine all terms containing n.

$$\left(-17\left(m^{2}+16\right)^{-\frac{1}{2}}m-17\left(m^{2}+16\right)^{-\frac{1}{2}}\right)n=4\sqrt{34}-68\left(m^{2}+16\right)^{-\frac{1}{2}}m-68\left(m^{2}+16\right)^{-\frac{1}{2}}$$

The equation is in standard form.

$$\frac{-17m-17}{\sqrt{m^{2}+16}}n=-\frac{68m}{\sqrt{m^{2}+16}}+4\sqrt{34}-\frac{68}{\sqrt{m^{2}+16}}$$

Divide both sides by -17\left(m^{2}+16\right)^{-\frac{1}{2}}m-17\left(m^{2}+16\right)^{-\frac{1}{2}}.

$$\frac{\frac{-17m-17}{\sqrt{m^{2}+16}}n\sqrt{m^{2}+16}}{-17m-17}=\frac{4\left(\sqrt{34m^{2}+544}-17m-17\right)}{\sqrt{m^{2}+16}\times \frac{-17m-17}{\sqrt{m^{2}+16}}}$$

Dividing by -17\left(m^{2}+16\right)^{-\frac{1}{2}}m-17\left(m^{2}+16\right)^{-\frac{1}{2}} undoes the multiplication by -17\left(m^{2}+16\right)^{-\frac{1}{2}}m-17\left(m^{2}+16\right)^{-\frac{1}{2}}.

$$n=\frac{4\left(\sqrt{34m^{2}+544}-17m-17\right)}{\sqrt{m^{2}+16}\times \frac{-17m-17}{\sqrt{m^{2}+16}}}$$

Divide \frac{4\left(\sqrt{34m^{2}+544}-17m-17\right)}{\sqrt{m^{2}+16}} by -17\left(m^{2}+16\right)^{-\frac{1}{2}}m-17\left(m^{2}+16\right)^{-\frac{1}{2}}.

$$n=-\frac{4\left(\sqrt{34m^{2}+544}-17m-17\right)}{17\left(m+1\right)}$$