Solve TRACB : x/y * a/b = 1 where x/y and a/b are rational numbers then the multiplicative invers | 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

$$\frac { x } { y } \times \frac { a } { b } = 1 \frac { x } { y } \frac { a } { b }$$

Solve for a (complex solution)

$a\in \mathrm{C}$
$y\neq 0\text{ and }b\neq 0$

Solution Steps

Multiply both sides of the equation by $by$, the least common multiple of $y,b$.

$$bx\times \frac{a}{b}=1bx\times \frac{a}{b}$$

Express $b\times \frac{a}{b}$ as a single fraction.

$$\frac{ba}{b}x=1bx\times \frac{a}{b}$$

Cancel out $b$ in both numerator and denominator.

$$ax=1bx\times \frac{a}{b}$$

Express $1\times \frac{a}{b}$ as a single fraction.

$$ax=\frac{a}{b}bx$$

Express $\frac{a}{b}b$ as a single fraction.

$$ax=\frac{ab}{b}x$$

Cancel out $b$ in both numerator and denominator.

$$ax=ax$$

Subtract $ax$ from both sides.

$$ax-ax=0$$

Combine $ax$ and $-ax$ to get $0$.

$$0=0$$

Compare $0$ and $0$.

$$\text{true}$$

This is true for any $a$.

$$a\in \mathrm{C}$$

Solve for a

$a\in \mathrm{R}$
$y\neq 0\text{ and }b\neq 0$

Solution Steps

Multiply both sides of the equation by $by$, the least common multiple of $y,b$.

$$bx\times \frac{a}{b}=1bx\times \frac{a}{b}$$

Express $b\times \frac{a}{b}$ as a single fraction.

$$\frac{ba}{b}x=1bx\times \frac{a}{b}$$

Cancel out $b$ in both numerator and denominator.

$$ax=1bx\times \frac{a}{b}$$

Express $1\times \frac{a}{b}$ as a single fraction.

$$ax=\frac{a}{b}bx$$

Express $\frac{a}{b}b$ as a single fraction.

$$ax=\frac{ab}{b}x$$

Cancel out $b$ in both numerator and denominator.

$$ax=ax$$

Subtract $ax$ from both sides.

$$ax-ax=0$$

Combine $ax$ and $-ax$ to get $0$.

$$0=0$$

Compare $0$ and $0$.

$$\text{true}$$

This is true for any $a$.

$$a\in \mathrm{R}$$

Solve for b

$b\neq 0$
$y\neq 0$

Short Solution Steps

Variable $b$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $by$, the least common multiple of $y,b$.

$$bx\times \frac{a}{b}=1bx\times \frac{a}{b}$$

Express $b\times \frac{a}{b}$ as a single fraction.

$$\frac{ba}{b}x=1bx\times \frac{a}{b}$$

Express $1\times \frac{a}{b}$ as a single fraction.

$$\frac{ba}{b}x=\frac{a}{b}bx$$

Express $\frac{ba}{b}x$ as a single fraction.

$$\frac{bax}{b}=\frac{a}{b}bx$$

Express $\frac{a}{b}b$ as a single fraction.

$$\frac{bax}{b}=\frac{ab}{b}x$$

Express $\frac{ab}{b}x$ as a single fraction.

$$\frac{bax}{b}=\frac{abx}{b}$$

Subtract $\frac{abx}{b}$ from both sides.

$$\frac{bax}{b}-\frac{abx}{b}=0$$

Since $\frac{bax}{b}$ and $\frac{abx}{b}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{bax-abx}{b}=0$$

Combine like terms in $bax-abx$.

$$\frac{0}{b}=0$$

Variable $b$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $b$.

$$0=0$$

Reorder the terms.

$$\text{true}$$

This is true for any $b$.

$$b\in \mathrm{R}$$

Variable $b$ cannot be equal to $0$.

$$b\in \mathrm{R}\setminus 0$$