Precalculus Examples

$x^{2} - y = 7$ , $2 x^{2} + 3 y = 24$

Step 1

Solve for $y$ in $x^{2} - y = 7$ .

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Step 1.1

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

$- y = 7 - x^{2}$

$2 x^{2} + 3 y = 24$

Step 1.2

Divide each term in $- y = 7 - x^{2}$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y = 7 - x^{2}$ by $- 1$ .

$\frac{- y}{- 1} = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$2 x^{2} + 3 y = 24$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$2 x^{2} + 3 y = 24$

Step 1.2.2.2

Divide $y$ by $1$ .

$y = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$2 x^{2} + 3 y = 24$

$y = \frac{7}{- 1} + \frac{- x^{2}}{- 1}$

$2 x^{2} + 3 y = 24$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $7$ by $- 1$ .

$y = - 7 + \frac{- x^{2}}{- 1}$

$2 x^{2} + 3 y = 24$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = - 7 + \frac{x^{2}}{1}$

$2 x^{2} + 3 y = 24$

Step 1.2.3.1.3

Divide $x^{2}$ by $1$ .

$y = - 7 + x^{2}$

$2 x^{2} + 3 y = 24$

$y = - 7 + x^{2}$

$2 x^{2} + 3 y = 24$

$y = - 7 + x^{2}$

$2 x^{2} + 3 y = 24$

$y = - 7 + x^{2}$

$2 x^{2} + 3 y = 24$

$y = - 7 + x^{2}$

$2 x^{2} + 3 y = 24$

Step 2

Replace all occurrences of $y$ with $- 7 + x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x^{2} + 3 y = 24$ with $- 7 + x^{2}$ .

$2 x^{2} + 3 (- 7 + x^{2}) = 24$

$y = - 7 + x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x^{2} + 3 (- 7 + x^{2})$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$2 x^{2} + 3 \cdot - 7 + 3 x^{2} = 24$

$y = - 7 + x^{2}$

Step 2.2.1.1.2

Multiply $3$ by $- 7$ .

$2 x^{2} - 21 + 3 x^{2} = 24$

$y = - 7 + x^{2}$

$2 x^{2} - 21 + 3 x^{2} = 24$

$y = - 7 + x^{2}$

Step 2.2.1.2

Add $2 x^{2}$ and $3 x^{2}$ .

$5 x^{2} - 21 = 24$

$y = - 7 + x^{2}$

$5 x^{2} - 21 = 24$

$y = - 7 + x^{2}$

$5 x^{2} - 21 = 24$

$y = - 7 + x^{2}$

$5 x^{2} - 21 = 24$

$y = - 7 + x^{2}$

Step 3

Solve for $x$ in $5 x^{2} - 21 = 24$ .

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Step 3.1

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

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Step 3.1.1

Add $21$ to both sides of the equation.

$5 x^{2} = 24 + 21$

$y = - 7 + x^{2}$

Step 3.1.2

Add $24$ and $21$ .

$5 x^{2} = 45$

$y = - 7 + x^{2}$

$5 x^{2} = 45$

$y = - 7 + x^{2}$

Step 3.2

Divide each term in $5 x^{2} = 45$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 x^{2} = 45$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{45}{5}$

$y = - 7 + x^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{45}{5}$

$y = - 7 + x^{2}$

Step 3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{45}{5}$

$y = - 7 + x^{2}$

$x^{2} = \frac{45}{5}$

$y = - 7 + x^{2}$

$x^{2} = \frac{45}{5}$

$y = - 7 + x^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $45$ by $5$ .

$x^{2} = 9$

$y = - 7 + x^{2}$

$x^{2} = 9$

$y = - 7 + x^{2}$

$x^{2} = 9$

$y = - 7 + x^{2}$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

$y = - 7 + x^{2}$

Step 3.4

Simplify $\pm \sqrt{9}$ .

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Step 3.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

$y = - 7 + x^{2}$

Step 3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

$y = - 7 + x^{2}$

$x = \pm 3$

$y = - 7 + x^{2}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

$y = - 7 + x^{2}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

$y = - 7 + x^{2}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

$y = - 7 + x^{2}$

$x = 3, - 3$

$y = - 7 + x^{2}$

$x = 3, - 3$

$y = - 7 + x^{2}$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $3$ .

$y = - 7 + {(3)}^{2}$

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 7 + {(3)}^{2}$ .

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Step 4.2.1.1

Raise $3$ to the power of $2$ .

$y = - 7 + 9$

$x = 3$

Step 4.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

Step 5

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

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Step 5.1

Replace all occurrences of $x$ in $y = - 7 + x^{2}$ with $- 3$ .

$y = - 7 + {(- 3)}^{2}$

$x = - 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 7 + {(- 3)}^{2}$ .

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Step 5.2.1.1

Raise $- 3$ to the power of $2$ .

$y = - 7 + 9$

$x = - 3$

Step 5.2.1.2

Add $- 7$ and $9$ .

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

$y = 2$

$x = - 3$

Step 6

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

$(3, 2)$

$(- 3, 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(3, 2), (- 3, 2)$

Equation Form:

$x = 3, y = 2$

$x = - 3, y = 2$

Step 8