Precalculus Examples

$y = x + 3$ , $4 x^{2} + y^{2} = 9$

Step 1

Replace all occurrences of $y$ with $x + 3$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $4 x^{2} + y^{2} = 9$ with $x + 3$ .

$4 x^{2} + {(x + 3)}^{2} = 9$

$y = x + 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $4 x^{2} + {(x + 3)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$4 x^{2} + (x + 3) (x + 3) = 9$

$y = x + 3$

Step 1.2.1.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$4 x^{2} + x (x + 3) + 3 (x + 3) = 9$

$y = x + 3$

Step 1.2.1.1.2.2

Apply the distributive property.

$4 x^{2} + x \cdot x + x \cdot 3 + 3 (x + 3) = 9$

$y = x + 3$

Step 1.2.1.1.2.3

Apply the distributive property.

$4 x^{2} + x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 = 9$

$y = x + 3$

$4 x^{2} + x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 = 9$

$y = x + 3$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$4 x^{2} + x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 = 9$

$y = x + 3$

Step 1.2.1.1.3.1.2

Move $3$ to the left of $x$ .

$4 x^{2} + x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 = 9$

$y = x + 3$

Step 1.2.1.1.3.1.3

Multiply $3$ by $3$ .

$4 x^{2} + x^{2} + 3 x + 3 x + 9 = 9$

$y = x + 3$

$4 x^{2} + x^{2} + 3 x + 3 x + 9 = 9$

$y = x + 3$

Step 1.2.1.1.3.2

Add $3 x$ and $3 x$ .

$4 x^{2} + x^{2} + 6 x + 9 = 9$

$y = x + 3$

$4 x^{2} + x^{2} + 6 x + 9 = 9$

$y = x + 3$

$4 x^{2} + x^{2} + 6 x + 9 = 9$

$y = x + 3$

Step 1.2.1.2

Add $4 x^{2}$ and $x^{2}$ .

$5 x^{2} + 6 x + 9 = 9$

$y = x + 3$

$5 x^{2} + 6 x + 9 = 9$

$y = x + 3$

$5 x^{2} + 6 x + 9 = 9$

$y = x + 3$

$5 x^{2} + 6 x + 9 = 9$

$y = x + 3$

Step 2

Solve for $x$ in $5 x^{2} + 6 x + 9 = 9$ .

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Step 2.1

Subtract $9$ from both sides of the equation.

$5 x^{2} + 6 x + 9 - 9 = 0$

$y = x + 3$

Step 2.2

Combine the opposite terms in $5 x^{2} + 6 x + 9 - 9$ .

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Step 2.2.1

Subtract $9$ from $9$ .

$5 x^{2} + 6 x + 0 = 0$

$y = x + 3$

Step 2.2.2

Add $5 x^{2} + 6 x$ and $0$ .

$5 x^{2} + 6 x = 0$

$y = x + 3$

$5 x^{2} + 6 x = 0$

$y = x + 3$

Step 2.3

Factor $x$ out of $5 x^{2} + 6 x$ .

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Step 2.3.1

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

$x (5 x) + 6 x = 0$

$y = x + 3$

Step 2.3.2

Factor $x$ out of $6 x$ .

$x (5 x) + x \cdot 6 = 0$

$y = x + 3$

Step 2.3.3

Factor $x$ out of $x (5 x) + x \cdot 6$ .

$x (5 x + 6) = 0$

$y = x + 3$

$x (5 x + 6) = 0$

$y = x + 3$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$5 x + 6 = 0$

$y = x + 3$

Step 2.5

Set $x$ equal to $0$ .

$x = 0$

$y = x + 3$

Step 2.6

Set $5 x + 6$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $5 x + 6$ equal to $0$ .

$5 x + 6 = 0$

$y = x + 3$

Step 2.6.2

Solve $5 x + 6 = 0$ for $x$ .

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Step 2.6.2.1

Subtract $6$ from both sides of the equation.

$5 x = - 6$

$y = x + 3$

Step 2.6.2.2

Divide each term in $5 x = - 6$ by $5$ and simplify.

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Step 2.6.2.2.1

Divide each term in $5 x = - 6$ by $5$ .

$\frac{5 x}{5} = \frac{- 6}{5}$

$y = x + 3$

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 6}{5}$

$y = x + 3$

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{5}$

$y = x + 3$

$x = \frac{- 6}{5}$

$y = x + 3$

$x = \frac{- 6}{5}$

$y = x + 3$

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{5}$

$y = x + 3$

$x = - \frac{6}{5}$

$y = x + 3$

$x = - \frac{6}{5}$

$y = x + 3$

$x = - \frac{6}{5}$

$y = x + 3$

$x = - \frac{6}{5}$

$y = x + 3$

Step 2.7

The final solution is all the values that make $x (5 x + 6) = 0$ true.

$x = 0, - \frac{6}{5}$

$y = x + 3$

$x = 0, - \frac{6}{5}$

$y = x + 3$

Step 3

Replace all occurrences of $x$ with $0$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x + 3$ with $0$ .

$y = (0) + 3$

$x = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Add $0$ and $3$ .

$y = 3$

$x = 0$

$y = 3$

$x = 0$

$y = 3$

$x = 0$

Step 4

Replace all occurrences of $x$ with $- \frac{6}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x + 3$ with $- \frac{6}{5}$ .

$y = (- \frac{6}{5}) + 3$

$x = - \frac{6}{5}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $(- \frac{6}{5}) + 3$ .

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Step 4.2.1.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{6}{5} + 3 \cdot \frac{5}{5}$

$x = - \frac{6}{5}$

Step 4.2.1.2

Combine $3$ and $\frac{5}{5}$ .

$y = - \frac{6}{5} + \frac{3 \cdot 5}{5}$

$x = - \frac{6}{5}$

Step 4.2.1.3

Combine the numerators over the common denominator.

$y = \frac{- 6 + 3 \cdot 5}{5}$

$x = - \frac{6}{5}$

Step 4.2.1.4

Simplify the numerator.

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Step 4.2.1.4.1

Multiply $3$ by $5$ .

$y = \frac{- 6 + 15}{5}$

$x = - \frac{6}{5}$

Step 4.2.1.4.2

Add $- 6$ and $15$ .

$y = \frac{9}{5}$

$x = - \frac{6}{5}$

$y = \frac{9}{5}$

$x = - \frac{6}{5}$

$y = \frac{9}{5}$

$x = - \frac{6}{5}$

$y = \frac{9}{5}$

$x = - \frac{6}{5}$

$y = \frac{9}{5}$

$x = - \frac{6}{5}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 3)$

$(- \frac{6}{5}, \frac{9}{5})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, 3), (- \frac{6}{5}, \frac{9}{5})$

Equation Form:

$x = 0, y = 3$

$x = - \frac{6}{5}, y = \frac{9}{5}$

Step 7