Solve for b a÷b=b÷a

$a \div b = b \div a$

Rewrite the division as a fraction.

$\frac{a}{b} = b \div a$

Rewrite the division as a fraction.

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

Subtract $\frac{b}{a}$ from both sides of the equation.

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

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$b, a, 1$

Since $b, a, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1$ then find LCM for the variable part $b^{1}, a^{1}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $b^{1}$ is $b$ itself.

$b^{1} = b$

$b$ occurs $1$ time.

The factor for $a^{1}$ is $a$ itself.

$a^{1} = a$

$a$ occurs $1$ time.

The LCM of $b^{1}, a^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$b \cdot a$

Multiply $b$ by $a$ .

$b a$

Multiply each term by $b a$ and simplify.

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Multiply each term in $\frac{a}{b} - \frac{b}{a} = 0$ by $b a$ in order to remove all the denominators from the equation.

$\frac{a}{b} \cdot (b a) - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Simplify each term.

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Cancel the common factor of $b$ .

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Factor $b$ out of $b a$ .

$\frac{a}{b} \cdot (b (a)) - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Cancel the common factor.

$\frac{a}{b} \cdot (b a) - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Rewrite the expression.

$a \cdot a - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Raise $a$ to the power of $1$ .

$a^{1} \cdot a - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Raise $a$ to the power of $1$ .

$a^{1} \cdot a^{1} - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{1 + 1} - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Add $1$ and $1$ .

$a^{2} - \frac{b}{a} \cdot (b a) = 0 \cdot (b a)$

Cancel the common factor of $a$ .

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Move the leading negative in $- \frac{b}{a}$ into the numerator.

$a^{2} + \frac{- b}{a} \cdot (b a) = 0 \cdot (b a)$

Factor $a$ out of $b a$ .

$a^{2} + \frac{- b}{a} \cdot (a b) = 0 \cdot (b a)$

Cancel the common factor.

$a^{2} + \frac{- b}{a} \cdot (a b) = 0 \cdot (b a)$

Rewrite the expression.

$a^{2} - b \cdot b = 0 \cdot (b a)$

Raise $b$ to the power of $1$ .

$a^{2} - (b^{1} b) = 0 \cdot (b a)$

Raise $b$ to the power of $1$ .

$a^{2} - (b^{1} b^{1}) = 0 \cdot (b a)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{2} - b^{1 + 1} = 0 \cdot (b a)$

Add $1$ and $1$ .

$a^{2} - b^{2} = 0 \cdot (b a)$

Multiply $0 (b a)$ .

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Multiply $b$ by $0$ .

$a^{2} - b^{2} = 0 a$

Multiply $0$ by $a$ .

$a^{2} - b^{2} = 0$

Solve the equation.

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Subtract $a^{2}$ from both sides of the equation.

$- b^{2} = - a^{2}$

Multiply each term in $- b^{2} = - a^{2}$ by $- 1$

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Multiply each term in $- b^{2} = - a^{2}$ by $- 1$ .

$(- b^{2}) \cdot - 1 = (- a^{2}) \cdot - 1$

Multiply $(- b^{2}) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 b^{2} = (- a^{2}) \cdot - 1$

Multiply $b^{2}$ by $1$ .

$b^{2} = (- a^{2}) \cdot - 1$

Multiply $(- a^{2}) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$b^{2} = 1 a^{2}$

Multiply $a^{2}$ by $1$ .

$b^{2} = a^{2}$

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| b | = | a |$

Solve for $b$ .

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Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$b = \pm a$

Set up the positive portion of the $\pm$ solution.

$b = a$

Set up the negative portion of the $\pm$ solution.

$b = - a$

The solution to the equation includes both the positive and negative portions of the solution.

$b = a$

$b = - a$

$b = a$

$b = - a$

$b = a$

$b = - a$

Solve for b a÷b=b÷a

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