Precalculus Examples

$x^{2} = 2 y + 10$ , $3 x - y = 9$

Step 1

Solve for $x$ in $3 x - y = 9$ .

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Step 1.1

Add $y$ to both sides of the equation.

$3 x = 9 + y$

$x^{2} = 2 y + 10$

Step 1.2

Divide each term in $3 x = 9 + y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 9 + y$ by $3$ .

$\frac{3 x}{3} = \frac{9}{3} + \frac{y}{3}$

$x^{2} = 2 y + 10$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{9}{3} + \frac{y}{3}$

$x^{2} = 2 y + 10$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{3} + \frac{y}{3}$

$x^{2} = 2 y + 10$

$x = \frac{9}{3} + \frac{y}{3}$

$x^{2} = 2 y + 10$

$x = \frac{9}{3} + \frac{y}{3}$

$x^{2} = 2 y + 10$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $9$ by $3$ .

$x = 3 + \frac{y}{3}$

$x^{2} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$x^{2} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$x^{2} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$x^{2} = 2 y + 10$

Step 2

Replace all occurrences of $x$ with $3 + \frac{y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} = 2 y + 10$ with $3 + \frac{y}{3}$ .

${(3 + \frac{y}{3})}^{2} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(3 + \frac{y}{3})}^{2}$ .

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Step 2.2.1.1

Rewrite ${(3 + \frac{y}{3})}^{2}$ as $(3 + \frac{y}{3}) (3 + \frac{y}{3})$ .

$(3 + \frac{y}{3}) (3 + \frac{y}{3}) = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.2

Expand $(3 + \frac{y}{3}) (3 + \frac{y}{3})$ using the FOIL Method.

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Step 2.2.1.2.1

Apply the distributive property.

$3 (3 + \frac{y}{3}) + \frac{y}{3} \cdot (3 + \frac{y}{3}) = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.2.2

Apply the distributive property.

$3 \cdot 3 + 3 (\frac{y}{3}) + \frac{y}{3} \cdot (3 + \frac{y}{3}) = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.2.3

Apply the distributive property.

$3 \cdot 3 + 3 (\frac{y}{3}) + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$3 \cdot 3 + 3 (\frac{y}{3}) + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3

Simplify and combine like terms.

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Step 2.2.1.3.1

Simplify each term.

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Step 2.2.1.3.1.1

Multiply $3$ by $3$ .

$9 + 3 (\frac{y}{3}) + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.3.1.2.1

Cancel the common factor.

$9 + 3 (\frac{y}{3}) + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.2.2

Rewrite the expression.

$9 + y + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$9 + y + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.3.1.3.1

Cancel the common factor.

$9 + y + \frac{y}{3} \cdot 3 + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.3.2

Rewrite the expression.

$9 + y + y + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$9 + y + y + \frac{y}{3} \cdot \frac{y}{3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.4

Multiply $\frac{y}{3} \cdot \frac{y}{3}$ .

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Step 2.2.1.3.1.4.1

Multiply $\frac{y}{3}$ by $\frac{y}{3}$ .

$9 + y + y + \frac{y \cdot y}{3 \cdot 3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.4.2

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

$9 + y + y + \frac{y \cdot y}{3 \cdot 3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.4.3

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

$9 + y + y + \frac{y \cdot y}{3 \cdot 3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.4.4

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

$9 + y + y + \frac{y^{1 + 1}}{3 \cdot 3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.4.5

Add $1$ and $1$ .

$9 + y + y + \frac{y^{2}}{3 \cdot 3} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.1.4.6

Multiply $3$ by $3$ .

$9 + y + y + \frac{y^{2}}{9} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$9 + y + y + \frac{y^{2}}{9} = 2 y + 10$

$x = 3 + \frac{y}{3}$

$9 + y + y + \frac{y^{2}}{9} = 2 y + 10$

$x = 3 + \frac{y}{3}$

Step 2.2.1.3.2

Add $y$ and $y$ .