Solve 19(2a_{1}+18d)=30(2a_{1}+2ad)

$$19 ( 2 a _ { 1 } + 18 d ) = 30 ( 2 a _ { 1 } + 2 a d )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{11a_{1}}{30d}+\frac{57}{10}\text{, }&d\neq 0\\a\in \mathrm{C}\text{, }&a_{1}=0\text{ and }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 19 by 2a_{1}+18d.

$$38a_{1}+342d=30\left(2a_{1}+2ad\right)$$

Use the distributive property to multiply 30 by 2a_{1}+2ad.

$$38a_{1}+342d=60a_{1}+60ad$$

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

$$60a_{1}+60ad=38a_{1}+342d$$

Subtract 60a_{1} from both sides.

$$60ad=38a_{1}+342d-60a_{1}$$

Combine 38a_{1} and -60a_{1} to get -22a_{1}.

$$60ad=-22a_{1}+342d$$

The equation is in standard form.

$$60da=342d-22a_{1}$$

Divide both sides by 60d.

$$\frac{60da}{60d}=\frac{342d-22a_{1}}{60d}$$

Dividing by 60d undoes the multiplication by 60d.

$$a=\frac{342d-22a_{1}}{60d}$$

Divide -22a_{1}+342d by 60d.

$$a=-\frac{11a_{1}}{30d}+\frac{57}{10}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{11a_{1}}{30d}+\frac{57}{10}\text{, }&d\neq 0\\a\in \mathrm{R}\text{, }&a_{1}=0\text{ and }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 19 by 2a_{1}+18d.

$$38a_{1}+342d=30\left(2a_{1}+2ad\right)$$

Use the distributive property to multiply 30 by 2a_{1}+2ad.

$$38a_{1}+342d=60a_{1}+60ad$$

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

$$60a_{1}+60ad=38a_{1}+342d$$

Subtract 60a_{1} from both sides.

$$60ad=38a_{1}+342d-60a_{1}$$

Combine 38a_{1} and -60a_{1} to get -22a_{1}.

$$60ad=-22a_{1}+342d$$

The equation is in standard form.

$$60da=342d-22a_{1}$$

Divide both sides by 60d.

$$\frac{60da}{60d}=\frac{342d-22a_{1}}{60d}$$

Dividing by 60d undoes the multiplication by 60d.

$$a=\frac{342d-22a_{1}}{60d}$$

Divide -22a_{1}+342d by 60d.

$$a=-\frac{11a_{1}}{30d}+\frac{57}{10}$$

Solve for a_1

$$a_{1}=\frac{3d\left(57-10a\right)}{11}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 19 by 2a_{1}+18d.

$$38a_{1}+342d=30\left(2a_{1}+2ad\right)$$

Use the distributive property to multiply 30 by 2a_{1}+2ad.

$$38a_{1}+342d=60a_{1}+60ad$$

Subtract 60a_{1} from both sides.

$$38a_{1}+342d-60a_{1}=60ad$$

Combine 38a_{1} and -60a_{1} to get -22a_{1}.

$$-22a_{1}+342d=60ad$$

Subtract 342d from both sides.

$$-22a_{1}=60ad-342d$$

Divide both sides by -22.

$$\frac{-22a_{1}}{-22}=\frac{6d\left(10a-57\right)}{-22}$$

Dividing by -22 undoes the multiplication by -22.

$$a_{1}=\frac{6d\left(10a-57\right)}{-22}$$

Divide 6d\left(-57+10a\right) by -22.

$$a_{1}=-\frac{3d\left(10a-57\right)}{11}$$