Solve p/(x + r) + q/(x - r) = (k/(2x))! | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{p}{x+r}+\frac{q}{x-r}=\frac{k}{2x}!$$

Solve for p

$p=-\frac{qr+qx+\left(\frac{k}{2x}\right)!r^{2}-\left(\frac{k}{2x}\right)!x^{2}}{x-r}$
$x\neq 0\text{ and }|x|\neq |r|$

Steps for Solving Linear Equation

Multiply both sides of the equation by $\left(x-r\right)\left(-x-r\right)$, the least common multiple of $x+r,x-r$.

$$\left(-x+r\right)p+\left(-x-r\right)q=\left(x-r\right)\left(-x-r\right)\times \left(\frac{k}{2x}\right)!$$

Use the distributive property to multiply $-x+r$ by $p$.

$$-xp+rp+\left(-x-r\right)q=\left(x-r\right)\left(-x-r\right)\times \left(\frac{k}{2x}\right)!$$

Use the distributive property to multiply $-x-r$ by $q$.

$$-xp+rp-xq-rq=\left(x-r\right)\left(-x-r\right)\times \left(\frac{k}{2x}\right)!$$

Use the distributive property to multiply $x-r$ by $-x-r$ and combine like terms.

$$-xp+rp-xq-rq=\left(-x^{2}+r^{2}\right)\times \left(\frac{k}{2x}\right)!$$

Use the distributive property to multiply $-x^{2}+r^{2}$ by $\left(\frac{k}{2x}\right)!$.

$$-xp+rp-xq-rq=-x^{2}\times \left(\frac{k}{2x}\right)!+r^{2}\times \left(\frac{k}{2x}\right)!$$

Add $xq$ to both sides.

$$-xp+rp-rq=-x^{2}\times \left(\frac{k}{2x}\right)!+r^{2}\times \left(\frac{k}{2x}\right)!+xq$$

Add $rq$ to both sides.

$$-xp+rp=-x^{2}\times \left(\frac{k}{2x}\right)!+r^{2}\times \left(\frac{k}{2x}\right)!+xq+rq$$

Combine all terms containing $p$.

$$\left(-x+r\right)p=-x^{2}\times \left(\frac{k}{2x}\right)!+r^{2}\times \left(\frac{k}{2x}\right)!+xq+rq$$

The equation is in standard form.

$$\left(r-x\right)p=qr+qx+\left(\frac{k}{2x}\right)!r^{2}-\left(\frac{k}{2x}\right)!x^{2}$$

Divide both sides by $-x+r$.

$$\frac{\left(r-x\right)p}{r-x}=\frac{qr+qx+\left(\frac{k}{2x}\right)!r^{2}-\left(\frac{k}{2x}\right)!x^{2}}{r-x}$$

Dividing by $-x+r$ undoes the multiplication by $-x+r$.

$$p=\frac{qr+qx+\left(\frac{k}{2x}\right)!r^{2}-\left(\frac{k}{2x}\right)!x^{2}}{r-x}$$