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 $a$ 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 $35 x^{4} y^{b}$ from both sides of the equation.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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Step 3.8.1

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

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

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 $0$ .

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

Step 3.8.3

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

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

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 $0$ .

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

Step 3.8.5

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

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

Step 3.8.6

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

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

Step 3.8.7

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

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

Step 3.8.8

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

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

Step 3.8.9

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

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

Step 3.8.10

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

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

Step 3.8.11

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

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

Step 3.8.12

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Rewrite the expression.

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

Step 4.2.2

Cancel the common factor of $y^{2}$ .

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Divide $a$ by $1$ .

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

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 $x^{5}$ .

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Step 4.3.1.1.1

Cancel the common factor.

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

Step 4.3.1.1.2

Rewrite the expression.

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

Step 4.3.1.2

Cancel the common factor of $y^{2}$ .

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Step 4.3.1.2.1

Cancel the common factor.

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

Step 4.3.1.2.2

Divide $21$ by $1$ .

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

Step 4.3.1.3

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

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Step 4.3.1.3.1

Factor $x^{4}$ out of $35 x^{4} y^{3}$ .

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

Step 4.3.1.3.2

Cancel the common factors.

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Step 4.3.1.3.2.1

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

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

Step 4.3.1.3.2.2

Cancel the common factor.

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

Step 4.3.1.3.2.3

Rewrite the expression.

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

Step 4.3.1.4

Cancel the common factor of $y^{3}$ and $y^{2}$ .

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Step 4.3.1.4.1

Factor $y^{2}$ out of $35 y^{3}$ .

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

Step 4.3.1.4.2

Cancel the common factors.

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Step 4.3.1.4.2.1

Factor $y^{2}$ out of $x y^{2}$ .

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

Step 4.3.1.4.2.2

Cancel the common factor.

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

Step 4.3.1.4.2.3

Rewrite the expression.

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

Step 4.3.1.5

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

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Step 4.3.1.5.1

Factor $x^{4}$ out of $- 35 x^{4} y^{b}$ .

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

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 $x^{5} y^{2}$ .

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

Step 4.3.1.5.2.2

Cancel the common factor.

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

Step 4.3.1.5.2.3

Rewrite the expression.

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

Step 4.3.1.6

Cancel the common factor of $y^{b}$ and $y^{2}$ .

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Step 4.3.1.6.1

Factor $y^{2}$ out of $- 35 y^{b}$ .

$a = 21 + \frac{35 y}{x} + \frac{y^{2} (- 35 y^{b - 2})}{x y^{2}}$

Step 4.3.1.6.2

Cancel the common factors.

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Step 4.3.1.6.2.1

Factor $y^{2}$ out of $x y^{2}$ .

$a = 21 + \frac{35 y}{x} + \frac{y^{2} (- 35 y^{b - 2})}{y^{2} x}$

Step 4.3.1.6.2.2

Cancel the common factor.

$a = 21 + \frac{35 y}{x} + \frac{y^{2} (- 35 y^{b - 2})}{y^{2} x}$

Step 4.3.1.6.2.3

Rewrite the expression.

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

Step 4.3.1.7

Move the negative in front of the fraction.

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

Step 4.3.2

To write $21$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4.3.3

Combine the numerators over the common denominator.

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

Step 4.3.4

Factor $7$ out of $21 x - 35 y^{b - 2}$ .

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Step 4.3.4.1

Factor $7$ out of $21 x$ .

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

Step 4.3.4.2

Factor $7$ out of $- 35 y^{b - 2}$ .

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

Step 4.3.4.3

Factor $7$ out of $7 (3 x) + 7 (- 5 y^{b - 2})$ .

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

Step 4.3.5

Combine the numerators over the common denominator.

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

Step 4.3.6

Factor $7$ out of $7 (3 x - 5 y^{b - 2}) + 35 y$ .

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Step 4.3.6.1

Factor $7$ out of $35 y$ .

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

Step 4.3.6.2

Factor $7$ out of $7 (3 x - 5 y^{b - 2}) + 7 (5 y)$ .

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