Algebra Examples

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

Step 1

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

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Step 1.1

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

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

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

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

Step 1.2.1.1.2

Simplify.

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Step 1.2.1.1.2.1

Multiply $3$ by $- 1$ .

$3 x - x^{2} - 3 x + 7 = - 2$

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

Step 1.2.1.1.2.2

Multiply $- 1$ by $- 7$ .

$3 x - x^{2} - 3 x + 7 = - 2$

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

$3 x - x^{2} - 3 x + 7 = - 2$

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

$3 x - x^{2} - 3 x + 7 = - 2$

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

Step 1.2.1.2

Combine the opposite terms in $3 x - x^{2} - 3 x + 7$ .

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Step 1.2.1.2.1

Subtract $3 x$ from $3 x$ .

$- x^{2} + 0 + 7 = - 2$

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

Step 1.2.1.2.2

Add $- x^{2}$ and $0$ .

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

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

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

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

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

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

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

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

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

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Subtract $7$ from both sides of the equation.

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

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

Step 2.1.2

Subtract $7$ from $- 2$ .

$- x^{2} = - 9$

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

$- x^{2} = - 9$

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

Step 2.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

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

Step 2.2.2.2

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

$x^{2} = \frac{- 9}{- 1}$

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

$x^{2} = \frac{- 9}{- 1}$

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

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

$x^{2} = 9$

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

$x^{2} = 9$

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

Step 2.3

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

$x = \pm \sqrt{9}$

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

Step 2.4

Simplify $\pm \sqrt{9}$ .

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Step 2.4.1

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

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

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

Step 2.4.2

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

$x = \pm 3$

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

$x = \pm 3$

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

Step 2.5

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

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Step 2.5.1

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

$x = 3$

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

Step 2.5.2

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

$x = - 3$

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

Step 2.5.3

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

$x = 3, - 3$

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

$x = 3, - 3$

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

$x = 3, - 3$

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

Step 3

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

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Step 3.1

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

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

$x = 3$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

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

$y = 9 + 3 (3) - 7$

$x = 3$

Step 3.2.1.1.2

Multiply $3$ by $3$ .

$y = 9 + 9 - 7$

$x = 3$

$y = 9 + 9 - 7$

$x = 3$

Step 3.2.1.2

Simplify by adding and subtracting.

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Step 3.2.1.2.1

Add $9$ and $9$ .

$y = 18 - 7$

$x = 3$

Step 3.2.1.2.2

Subtract $7$ from $18$ .

$y = 11$

$x = 3$

$y = 11$

$x = 3$

$y = 11$

$x = 3$

$y = 11$

$x = 3$

$y = 11$

$x = 3$

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 $y = x^{2} + 3 x - 7$ with $- 3$ .

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

$x = - 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

$y = 9 + 3 (- 3) - 7$

$x = - 3$

Step 4.2.1.1.2

Multiply $3$ by $- 3$ .

$y = 9 - 9 - 7$

$x = - 3$

$y = 9 - 9 - 7$

$x = - 3$

Step 4.2.1.2

Simplify by subtracting numbers.

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Step 4.2.1.2.1

Subtract $9$ from $9$ .

$y = 0 - 7$

$x = - 3$

Step 4.2.1.2.2

Subtract $7$ from $0$ .

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

$y = - 7$

$x = - 3$

Step 5

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

$(3, 11)$

$(- 3, - 7)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, 11), (- 3, - 7)$

Equation Form:

$x = 3, y = 11$

$x = - 3, y = - 7$

Step 7