Solve frac{14left({a}_{1}+{a}_{1}+13dright)}{2}-frac{11left({a}_{1}+{a}_{1}+b{a}_{2}right)}{2}=19

$$\frac{ 14 \left( { a }_{ 1 } + { a }_{ 1 } +13d \right) }{ 2 } - \frac{ 11 \left( { a }_{ 1 } + { a }_{ 1 } +b { a }_{ 2 } \right) }{ 2 } = 19$$

Solve for a_2 (complex solution)

$$\left\{\begin{matrix}a_{2}=-\frac{2\left(19-91d-3a_{1}\right)}{11b}\text{, }&b\neq 0\\a_{2}\in \mathrm{C}\text{, }&a_{1}=\frac{19-91d}{3}\text{ and }b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$14\left(a_{1}+a_{1}+13d\right)-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Combine a_{1} and a_{1} to get 2a_{1}.

$$14\left(2a_{1}+13d\right)-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Use the distributive property to multiply 14 by 2a_{1}+13d.

$$28a_{1}+182d-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Combine a_{1} and a_{1} to get 2a_{1}.

$$28a_{1}+182d-11\left(2a_{1}+ba_{2}\right)=38$$

Use the distributive property to multiply -11 by 2a_{1}+ba_{2}.

$$28a_{1}+182d-22a_{1}-11ba_{2}=38$$

Combine 28a_{1} and -22a_{1} to get 6a_{1}.

$$6a_{1}+182d-11ba_{2}=38$$

Subtract 6a_{1} from both sides.

$$182d-11ba_{2}=38-6a_{1}$$

Subtract 182d from both sides.

$$-11ba_{2}=38-6a_{1}-182d$$

The equation is in standard form.

$$\left(-11b\right)a_{2}=38-182d-6a_{1}$$

Divide both sides by -11b.

$$\frac{\left(-11b\right)a_{2}}{-11b}=\frac{38-182d-6a_{1}}{-11b}$$

Dividing by -11b undoes the multiplication by -11b.

$$a_{2}=\frac{38-182d-6a_{1}}{-11b}$$

Divide 38-6a_{1}-182d by -11b.

$$a_{2}=-\frac{2\left(19-91d-3a_{1}\right)}{11b}$$

Solve for a_1

$$a_{1}=\frac{11a_{2}b}{6}-\frac{91d}{3}+\frac{19}{3}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$14\left(a_{1}+a_{1}+13d\right)-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Combine a_{1} and a_{1} to get 2a_{1}.

$$14\left(2a_{1}+13d\right)-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Use the distributive property to multiply 14 by 2a_{1}+13d.

$$28a_{1}+182d-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Combine a_{1} and a_{1} to get 2a_{1}.

$$28a_{1}+182d-11\left(2a_{1}+ba_{2}\right)=38$$

Use the distributive property to multiply -11 by 2a_{1}+ba_{2}.

$$28a_{1}+182d-22a_{1}-11ba_{2}=38$$

Combine 28a_{1} and -22a_{1} to get 6a_{1}.

$$6a_{1}+182d-11ba_{2}=38$$

Subtract 182d from both sides.

$$6a_{1}-11ba_{2}=38-182d$$

Add 11ba_{2} to both sides.

$$6a_{1}=38-182d+11ba_{2}$$

The equation is in standard form.

$$6a_{1}=11a_{2}b-182d+38$$

Divide both sides by 6.

$$\frac{6a_{1}}{6}=\frac{11a_{2}b-182d+38}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$a_{1}=\frac{11a_{2}b-182d+38}{6}$$

Divide 38-182d+11ba_{2} by 6.

$$a_{1}=\frac{11a_{2}b}{6}-\frac{91d}{3}+\frac{19}{3}$$

Solve for a_2

$$\left\{\begin{matrix}a_{2}=-\frac{2\left(19-91d-3a_{1}\right)}{11b}\text{, }&b\neq 0\\a_{2}\in \mathrm{R}\text{, }&a_{1}=\frac{19-91d}{3}\text{ and }b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$14\left(a_{1}+a_{1}+13d\right)-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Combine a_{1} and a_{1} to get 2a_{1}.

$$14\left(2a_{1}+13d\right)-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Use the distributive property to multiply 14 by 2a_{1}+13d.

$$28a_{1}+182d-11\left(a_{1}+a_{1}+ba_{2}\right)=38$$

Combine a_{1} and a_{1} to get 2a_{1}.

$$28a_{1}+182d-11\left(2a_{1}+ba_{2}\right)=38$$

Use the distributive property to multiply -11 by 2a_{1}+ba_{2}.

$$28a_{1}+182d-22a_{1}-11ba_{2}=38$$

Combine 28a_{1} and -22a_{1} to get 6a_{1}.

$$6a_{1}+182d-11ba_{2}=38$$

Subtract 6a_{1} from both sides.

$$182d-11ba_{2}=38-6a_{1}$$

Subtract 182d from both sides.

$$-11ba_{2}=38-6a_{1}-182d$$

The equation is in standard form.

$$\left(-11b\right)a_{2}=38-182d-6a_{1}$$

Divide both sides by -11b.

$$\frac{\left(-11b\right)a_{2}}{-11b}=\frac{38-182d-6a_{1}}{-11b}$$

Dividing by -11b undoes the multiplication by -11b.

$$a_{2}=\frac{38-182d-6a_{1}}{-11b}$$

Divide 38-6a_{1}-182d by -11b.

$$a_{2}=-\frac{2\left(19-91d-3a_{1}\right)}{11b}$$