Solve a/3=2/5xquad(5)quad(a+b)x/2

$$\frac { a } { 3 } = \frac { 2 } { 5 } x \quad ( 5 ) \quad \frac { ( a + b ) x } { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{3bx^{2}}{1-3x^{2}}\text{, }&x\neq -\frac{\sqrt{3}}{3}\text{ and }x\neq \frac{\sqrt{3}}{3}\\a\in \mathrm{C}\text{, }&\left(x=\frac{\sqrt{3}}{3}\text{ or }x=-\frac{\sqrt{3}}{3}\right)\text{ and }b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 3,5,2.

$$10a=6x\times 5\left(a+b\right)x$$

Multiply x and x to get x^{2}.

$$10a=6x^{2}\times 5\left(a+b\right)$$

Multiply 6 and 5 to get 30.

$$10a=30x^{2}\left(a+b\right)$$

Use the distributive property to multiply 30x^{2} by a+b.

$$10a=30x^{2}a+30x^{2}b$$

Subtract 30x^{2}a from both sides.

$$10a-30x^{2}a=30x^{2}b$$

Combine all terms containing a.

$$\left(10-30x^{2}\right)a=30x^{2}b$$

The equation is in standard form.

$$\left(10-30x^{2}\right)a=30bx^{2}$$

Divide both sides by 10-30x^{2}.

$$\frac{\left(10-30x^{2}\right)a}{10-30x^{2}}=\frac{30bx^{2}}{10-30x^{2}}$$

Dividing by 10-30x^{2} undoes the multiplication by 10-30x^{2}.

$$a=\frac{30bx^{2}}{10-30x^{2}}$$

Divide 30x^{2}b by 10-30x^{2}.

$$a=\frac{3bx^{2}}{1-3x^{2}}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-a+\frac{a}{3x^{2}}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&a=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 3,5,2.

$$10a=6x\times 5\left(a+b\right)x$$

Multiply x and x to get x^{2}.

$$10a=6x^{2}\times 5\left(a+b\right)$$

Multiply 6 and 5 to get 30.

$$10a=30x^{2}\left(a+b\right)$$

Use the distributive property to multiply 30x^{2} by a+b.

$$10a=30x^{2}a+30x^{2}b$$

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

$$30x^{2}a+30x^{2}b=10a$$

Subtract 30x^{2}a from both sides.

$$30x^{2}b=10a-30x^{2}a$$

The equation is in standard form.

$$30x^{2}b=10a-30ax^{2}$$

Divide both sides by 30x^{2}.

$$\frac{30x^{2}b}{30x^{2}}=\frac{10a\left(1-3x^{2}\right)}{30x^{2}}$$

Dividing by 30x^{2} undoes the multiplication by 30x^{2}.

$$b=\frac{10a\left(1-3x^{2}\right)}{30x^{2}}$$

Divide 10a\left(1-3x^{2}\right) by 30x^{2}.

$$b=-a+\frac{a}{3x^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{3bx^{2}}{1-3x^{2}}\text{, }&|x|\neq \frac{\sqrt{3}}{3}\\a\in \mathrm{R}\text{, }&b=0\text{ and }|x|=\frac{\sqrt{3}}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 3,5,2.

$$10a=6x\times 5\left(a+b\right)x$$

Multiply x and x to get x^{2}.

$$10a=6x^{2}\times 5\left(a+b\right)$$

Multiply 6 and 5 to get 30.

$$10a=30x^{2}\left(a+b\right)$$

Use the distributive property to multiply 30x^{2} by a+b.

$$10a=30x^{2}a+30x^{2}b$$

Subtract 30x^{2}a from both sides.

$$10a-30x^{2}a=30x^{2}b$$

Combine all terms containing a.

$$\left(10-30x^{2}\right)a=30x^{2}b$$

The equation is in standard form.

$$\left(10-30x^{2}\right)a=30bx^{2}$$

Divide both sides by 10-30x^{2}.

$$\frac{\left(10-30x^{2}\right)a}{10-30x^{2}}=\frac{30bx^{2}}{10-30x^{2}}$$

Dividing by 10-30x^{2} undoes the multiplication by 10-30x^{2}.

$$a=\frac{30bx^{2}}{10-30x^{2}}$$

Divide 30x^{2}b by 10-30x^{2}.

$$a=\frac{3bx^{2}}{1-3x^{2}}$$

Solve for b

$$\left\{\begin{matrix}b=-a+\frac{a}{3x^{2}}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&a=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30, the least common multiple of 3,5,2.

$$10a=6x\times 5\left(a+b\right)x$$

Multiply x and x to get x^{2}.

$$10a=6x^{2}\times 5\left(a+b\right)$$

Multiply 6 and 5 to get 30.

$$10a=30x^{2}\left(a+b\right)$$

Use the distributive property to multiply 30x^{2} by a+b.

$$10a=30x^{2}a+30x^{2}b$$

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

$$30x^{2}a+30x^{2}b=10a$$

Subtract 30x^{2}a from both sides.

$$30x^{2}b=10a-30x^{2}a$$

The equation is in standard form.

$$30x^{2}b=10a-30ax^{2}$$

Divide both sides by 30x^{2}.

$$\frac{30x^{2}b}{30x^{2}}=\frac{10a\left(1-3x^{2}\right)}{30x^{2}}$$

Dividing by 30x^{2} undoes the multiplication by 30x^{2}.

$$b=\frac{10a\left(1-3x^{2}\right)}{30x^{2}}$$

Divide 10a\left(1-3x^{2}\right) by 30x^{2}.

$$b=-a+\frac{a}{3x^{2}}$$