Algebra Examples

${(2 + y)}^{2} - y^{2} = - 5 x + 9$

Step 1

Solve the equation for $y$ .

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Step 1.1

Move all terms to the left side of the equation and simplify.

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Step 1.1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1.1

Add $5 x$ to both sides of the equation.

${(2 + y)}^{2} - y^{2} + 5 x = 9$

Step 1.1.1.2

Subtract $9$ from both sides of the equation.

${(2 + y)}^{2} - y^{2} + 5 x - 9 = 0$

Step 1.1.2

Simplify ${(2 + y)}^{2} - y^{2} + 5 x - 9$ .

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Rewrite ${(2 + y)}^{2}$ as $(2 + y) (2 + y)$ .

$(2 + y) (2 + y) - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.2

Expand $(2 + y) (2 + y)$ using the FOIL Method.

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Step 1.1.2.1.2.1

Apply the distributive property.

$2 (2 + y) + y (2 + y) - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.2.2

Apply the distributive property.

$2 \cdot 2 + 2 y + y (2 + y) - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.2.3

Apply the distributive property.

$2 \cdot 2 + 2 y + y \cdot 2 + y \cdot y - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.3

Simplify and combine like terms.

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Step 1.1.2.1.3.1

Simplify each term.

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Step 1.1.2.1.3.1.1

Multiply $2$ by $2$ .

$4 + 2 y + y \cdot 2 + y \cdot y - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.3.1.2

Move $2$ to the left of $y$ .

$4 + 2 y + 2 \cdot y + y \cdot y - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.3.1.3

Multiply $y$ by $y$ .

$4 + 2 y + 2 y + y^{2} - y^{2} + 5 x - 9 = 0$

Step 1.1.2.1.3.2

Add $2 y$ and $2 y$ .

$4 + 4 y + y^{2} - y^{2} + 5 x - 9 = 0$

Step 1.1.2.2

Combine the opposite terms in $4 + 4 y + y^{2} - y^{2} + 5 x - 9$ .

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Step 1.1.2.2.1

Subtract $y^{2}$ from $y^{2}$ .

$4 + 4 y + 0 + 5 x - 9 = 0$

Step 1.1.2.2.2

Add $4 + 4 y$ and $0$ .

$4 + 4 y + 5 x - 9 = 0$

Step 1.1.2.3

Subtract $9$ from $4$ .

$4 y + 5 x - 5 = 0$

Step 1.2

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

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Step 1.2.1

Subtract $5 x$ from both sides of the equation.

$4 y - 5 = - 5 x$

Step 1.2.2

Add $5$ to both sides of the equation.

$4 y = - 5 x + 5$

Step 1.3

Divide each term in $4 y = - 5 x + 5$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 y = - 5 x + 5$ by $4$ .

$\frac{4 y}{4} = \frac{- 5 x}{4} + \frac{5}{4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 5 x}{4} + \frac{5}{4}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5 x}{4} + \frac{5}{4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative in front of the fraction.

$y = - \frac{5 x}{4} + \frac{5}{4}$

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degree of variable $y$ is $1$ and the degree of variable $x$ is $1$ .

Linear