Solve a(q-r)+b(r-p)+d(p-q)=0

$$a ( q - r ) + b ( r - p ) + d ( p - q ) = 0$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{dp-bp-dq+br}{q-r}\text{, }&q\neq r\\a\in \mathrm{R}\text{, }&\left(q=p\text{ and }r=p\right)\text{ or }\left(q=r\text{ and }b=d\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by q-r.

$$aq-ar+b\left(r-p\right)+d\left(p-q\right)=0$$

Use the distributive property to multiply b by r-p.

$$aq-ar+br-bp+d\left(p-q\right)=0$$

Use the distributive property to multiply d by p-q.

$$aq-ar+br-bp+dp-dq=0$$

Subtract br from both sides. Anything subtracted from zero gives its negation.

$$aq-ar-bp+dp-dq=-br$$

Add bp to both sides.

$$aq-ar+dp-dq=-br+bp$$

Subtract dp from both sides.

$$aq-ar-dq=-br+bp-dp$$

Add dq to both sides.

$$aq-ar=-br+bp-dp+dq$$

Reorder the terms.

$$aq-ar=bp-dp+dq-br$$

Combine all terms containing a.

$$\left(q-r\right)a=bp-dp+dq-br$$

Divide both sides by q-r.

$$\frac{\left(q-r\right)a}{q-r}=\frac{bp-dp+dq-br}{q-r}$$

Dividing by q-r undoes the multiplication by q-r.

$$a=\frac{bp-dp+dq-br}{q-r}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{dp+aq-dq-ar}{r-p}\text{, }&r\neq p\\b\in \mathrm{R}\text{, }&\left(a=d\text{ or }q=p\right)\text{ and }r=p\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a by q-r.

$$aq-ar+b\left(r-p\right)+d\left(p-q\right)=0$$

Use the distributive property to multiply b by r-p.

$$aq-ar+br-bp+d\left(p-q\right)=0$$

Use the distributive property to multiply d by p-q.

$$aq-ar+br-bp+dp-dq=0$$

Subtract aq from both sides. Anything subtracted from zero gives its negation.

$$-ar+br-bp+dp-dq=-aq$$

Add ar to both sides.

$$br-bp+dp-dq=-aq+ar$$

Subtract dp from both sides.

$$br-bp-dq=-aq+ar-dp$$

Add dq to both sides.

$$br-bp=-aq+ar-dp+dq$$

Reorder the terms.

$$-bp+br=-dp+dq-aq+ar$$

Combine all terms containing b.

$$\left(-p+r\right)b=-dp+dq-aq+ar$$

The equation is in standard form.

$$\left(r-p\right)b=ar-aq+dq-dp$$

Divide both sides by r-p.

$$\frac{\left(r-p\right)b}{r-p}=\frac{ar-aq+dq-dp}{r-p}$$

Dividing by r-p undoes the multiplication by r-p.

$$b=\frac{ar-aq+dq-dp}{r-p}$$