Algebra Examples

${(x + y)}^{7} = x^{7} + 7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}$

Step 1

Rewrite the equation as $x^{7} + 7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = {(x + y)}^{7}$ .

$x^{7} + 7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = {(x + y)}^{7}$

Step 2

Simplify ${(x + y)}^{7}$ .

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Step 2.1

Rewrite.

$x^{7} + 7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = 0 + 0 + {(x + y)}^{7}$

Step 2.2

Simplify by adding zeros.

$x^{7} + 7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = {(x + y)}^{7}$

Step 2.3

Use the Binomial Theorem.

$x^{7} + 7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}$

Step 3

Move all terms not containing $b$ to the right side of the equation.

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Step 3.1

Subtract $x^{7}$ from both sides of the equation.

$7 x^{6} y + a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7}$

Step 3.2

Subtract $7 x^{6} y$ from both sides of the equation.

$a x^{5} y^{2} + 35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y$

Step 3.3

Subtract $a x^{5} y^{2}$ from both sides of the equation.

$35 x^{4} y^{b} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y - a x^{5} y^{2}$

Step 3.4

Subtract $35 x^{3} y^{4}$ from both sides of the equation.

$35 x^{4} y^{b} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4}$

Step 3.5

Subtract $21 x^{2} y^{5}$ from both sides of the equation.

$35 x^{4} y^{b} + 7 x y^{6} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5}$

Step 3.6

Subtract $7 x y^{6}$ from both sides of the equation.

$35 x^{4} y^{b} + y^{7} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6}$

Step 3.7

Subtract $y^{7}$ from both sides of the equation.

$35 x^{4} y^{b} = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8

Combine the opposite terms in $x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - x^{7} - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$ .

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Step 3.8.1

Subtract $x^{7}$ from $x^{7}$ .

$35 x^{4} y^{b} = 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} + 0 - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8.2

Add $7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}$ and $0$ .

$35 x^{4} y^{b} = 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - 7 x^{6} y - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8.3

Subtract $7 x^{6} y$ from $7 x^{6} y$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} + 0 - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8.4

Add $21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}$ and $0$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - a x^{5} y^{2} - 35 x^{3} y^{4} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8.5

Subtract $35 x^{3} y^{4}$ from $35 x^{3} y^{4}$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - a x^{5} y^{2} + 0 - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8.6

Add $21 x^{5} y^{2} + 35 x^{4} y^{3} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - a x^{5} y^{2}$ and $0$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} - a x^{5} y^{2} - 21 x^{2} y^{5} - 7 x y^{6} - y^{7}$

Step 3.8.7

Subtract $21 x^{2} y^{5}$ from $21 x^{2} y^{5}$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + 7 x y^{6} + y^{7} - a x^{5} y^{2} + 0 - 7 x y^{6} - y^{7}$

Step 3.8.8

Add $21 x^{5} y^{2} + 35 x^{4} y^{3} + 7 x y^{6} + y^{7} - a x^{5} y^{2}$ and $0$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + 7 x y^{6} + y^{7} - a x^{5} y^{2} - 7 x y^{6} - y^{7}$

Step 3.8.9

Subtract $7 x y^{6}$ from $7 x y^{6}$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + y^{7} - a x^{5} y^{2} + 0 - y^{7}$

Step 3.8.10

Add $21 x^{5} y^{2} + 35 x^{4} y^{3} + y^{7} - a x^{5} y^{2}$ and $0$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} + y^{7} - a x^{5} y^{2} - y^{7}$

Step 3.8.11

Subtract $y^{7}$ from $y^{7}$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} - a x^{5} y^{2} + 0$

Step 3.8.12

Add $21 x^{5} y^{2} + 35 x^{4} y^{3} - a x^{5} y^{2}$ and $0$ .

$35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} - a x^{5} y^{2}$

Step 4

Divide each term in $35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} - a x^{5} y^{2}$ by $35 x^{4}$ and simplify.

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Step 4.1

Divide each term in $35 x^{4} y^{b} = 21 x^{5} y^{2} + 35 x^{4} y^{3} - a x^{5} y^{2}$ by $35 x^{4}$ .

$\frac{35 x^{4} y^{b}}{35 x^{4}} = \frac{21 x^{5} y^{2}}{35 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $35$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{35 x^{4} y^{b}}{35 x^{4}} = \frac{21 x^{5} y^{2}}{35 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.2.1.2

Rewrite the expression.

$\frac{x^{4} y^{b}}{x^{4}} = \frac{21 x^{5} y^{2}}{35 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.2.2

Cancel the common factor of $x^{4}$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{x^{4} y^{b}}{x^{4}} = \frac{21 x^{5} y^{2}}{35 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.2.2.2

Divide $y^{b}$ by $1$ .

$y^{b} = \frac{21 x^{5} y^{2}}{35 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $21$ and $35$ .

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Step 4.3.1.1.1

Factor $7$ out of $21 x^{5} y^{2}$ .

$y^{b} = \frac{7 (3 x^{5} y^{2})}{35 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

Factor $7$ out of $35 x^{4}$ .

$y^{b} = \frac{7 (3 x^{5} y^{2})}{7 (5 x^{4})} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.1.2.2

Cancel the common factor.

$y^{b} = \frac{7 (3 x^{5} y^{2})}{7 (5 x^{4})} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.1.2.3

Rewrite the expression.

$y^{b} = \frac{3 x^{5} y^{2}}{5 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.2

Cancel the common factor of $x^{5}$ and $x^{4}$ .

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Step 4.3.1.2.1

Factor $x^{4}$ out of $3 x^{5} y^{2}$ .

$y^{b} = \frac{x^{4} (3 x y^{2})}{5 x^{4}} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.2.2

Cancel the common factors.

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Step 4.3.1.2.2.1

Factor $x^{4}$ out of $5 x^{4}$ .

$y^{b} = \frac{x^{4} (3 x y^{2})}{x^{4} \cdot 5} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.2.2.2

Cancel the common factor.

$y^{b} = \frac{x^{4} (3 x y^{2})}{x^{4} \cdot 5} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.2.2.3

Rewrite the expression.

$y^{b} = \frac{3 x y^{2}}{5} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.3

Cancel the common factor of $35$ .

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Step 4.3.1.3.1

Cancel the common factor.

$y^{b} = \frac{3 x y^{2}}{5} + \frac{35 x^{4} y^{3}}{35 x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.3.2

Rewrite the expression.

$y^{b} = \frac{3 x y^{2}}{5} + \frac{x^{4} y^{3}}{x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.4

Cancel the common factor of $x^{4}$ .

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Step 4.3.1.4.1

Cancel the common factor.

$y^{b} = \frac{3 x y^{2}}{5} + \frac{x^{4} y^{3}}{x^{4}} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.4.2

Divide $y^{3}$ by $1$ .

$y^{b} = \frac{3 x y^{2}}{5} + y^{3} + \frac{- a x^{5} y^{2}}{35 x^{4}}$

Step 4.3.1.5

Cancel the common factor of $x^{5}$ and $x^{4}$ .

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Step 4.3.1.5.1

Factor $x^{4}$ out of $- a x^{5} y^{2}$ .

$y^{b} = \frac{3 x y^{2}}{5} + y^{3} + \frac{x^{4} (- a x y^{2})}{35 x^{4}}$

Step 4.3.1.5.2

Cancel the common factors.

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Step 4.3.1.5.2.1

Factor $x^{4}$ out of $35 x^{4}$ .

$y^{b} = \frac{3 x y^{2}}{5} + y^{3} + \frac{x^{4} (- a x y^{2})}{x^{4} \cdot 35}$

Step 4.3.1.5.2.2

Cancel the common factor.

$y^{b} = \frac{3 x y^{2}}{5} + y^{3} + \frac{x^{4} (- a x y^{2})}{x^{4} \cdot 35}$

Step 4.3.1.5.2.3

Rewrite the expression.

$y^{b} = \frac{3 x y^{2}}{5} + y^{3} + \frac{- a x y^{2}}{35}$

Step 4.3.1.6

Move the negative in front of the fraction.

$y^{b} = \frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35}$

Step 5

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (y^{b}) = \ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})$

Step 6

Expand $\ln (y^{b})$ by moving $b$ outside the logarithm.

$b \ln (y) = \ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})$

Step 7

Divide each term in $b \ln (y) = \ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})$ by $\ln (y)$ and simplify.

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Step 7.1

Divide each term in $b \ln (y) = \ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})$ by $\ln (y)$ .

$\frac{b \ln (y)}{\ln (y)} = \frac{\ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})}{\ln (y)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (y)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{b \ln (y)}{\ln (y)} = \frac{\ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})}{\ln (y)}$

Step 7.2.1.2

Divide $b$ by $1$ .

$b = \frac{\ln (\frac{3 x y^{2}}{5} + y^{3} - \frac{a x y^{2}}{35})}{\ln (y)}$