Linear Algebra Examples

x^{2} - y = 2

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

2 x - y = - 1

$2 x - y = - 1$

Step 1

Solve for

y

$y$ in

x^{2} - y = 2

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

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Step 1.1

Subtract

x^{2}

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

- y = 2 - x^{2}

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

2 x - y = - 1

$2 x - y = - 1$

Step 1.2

Divide each term in

- y = 2 - x^{2}

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

- 1

$- 1$ and simplify.

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Step 1.2.1

Divide each term in

- y = 2 - x^{2}

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

- 1

$- 1$ .

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

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

2 x - y = - 1

$2 x - y = - 1$

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{2}{- 1} + \frac{- x^{2}}{- 1}

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

2 x - y = - 1

$2 x - y = - 1$

Step 1.2.2.2

Divide

y

$y$ by

1

$1$ .

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

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

2 x - y = - 1

$2 x - y = - 1$

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

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

2 x - y = - 1

$2 x - y = - 1$

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

2

$2$ by

- 1

$- 1$ .

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

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

2 x - y = - 1

$2 x - y = - 1$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

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

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

2 x - y = - 1

$2 x - y = - 1$

Step 1.2.3.1.3

Divide

x^{2}

$x^{2}$ by

1

$1$ .

y = - 2 + x^{2}

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

2 x - y = - 1

$2 x - y = - 1$

y = - 2 + x^{2}

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

2 x - y = - 1

$2 x - y = - 1$

y = - 2 + x^{2}

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

2 x - y = - 1

$2 x - y = - 1$

y = - 2 + x^{2}

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

2 x - y = - 1

$2 x - y = - 1$

y = - 2 + x^{2}

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

2 x - y = - 1

$2 x - y = - 1$

Step 2

Replace all occurrences of

y

$y$ with

- 2 + x^{2}

$- 2 + x^{2}$ in each equation.

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Step 2.1

Replace all occurrences of

y

$y$ in

2 x - y = - 1

$2 x - y = - 1$ with

- 2 + x^{2}

$- 2 + x^{2}$ .

2 x - (- 2 + x^{2}) = - 1

$2 x - (- 2 + x^{2}) = - 1$

y = - 2 + x^{2}

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

2 x + 2 - x^{2} = - 1

$2 x + 2 - x^{2} = - 1$

y = - 2 + x^{2}

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

Step 2.2.1.2

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

2 x + 2 - x^{2} = - 1

$2 x + 2 - x^{2} = - 1$

y = - 2 + x^{2}

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

2 x + 2 - x^{2} = - 1

$2 x + 2 - x^{2} = - 1$

y = - 2 + x^{2}

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

2 x + 2 - x^{2} = - 1

$2 x + 2 - x^{2} = - 1$

y = - 2 + x^{2}

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

2 x + 2 - x^{2} = - 1

$2 x + 2 - x^{2} = - 1$

y = - 2 + x^{2}

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

Step 3

Solve for

x

$x$ in

2 x + 2 - x^{2} = - 1

$2 x + 2 - x^{2} = - 1$ .

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Step 3.1

Add

1

$1$ to both sides of the equation.

2 x + 2 - x^{2} + 1 = 0

$2 x + 2 - x^{2} + 1 = 0$

y = - 2 + x^{2}

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

Step 3.2

Add

2

$2$ and

1

$1$ .

2 x - x^{2} + 3 = 0

$2 x - x^{2} + 3 = 0$

y = - 2 + x^{2}

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

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor

- 1

$- 1$ out of

2 x - x^{2} + 3

$2 x - x^{2} + 3$ .

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Step 3.3.1.1

Reorder

2 x

$2 x$ and

- x^{2}

$- x^{2}$ .

- x^{2} + 2 x + 3 = 0

$- x^{2} + 2 x + 3 = 0$

y = - 2 + x^{2}

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

Step 3.3.1.2

Factor

- 1

$- 1$ out of

- x^{2}

$- x^{2}$ .

- (x^{2}) + 2 x + 3 = 0

$- (x^{2}) + 2 x + 3 = 0$

y = - 2 + x^{2}

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

Step 3.3.1.3

Factor

- 1

$- 1$ out of

2 x

$2 x$ .

- (x^{2}) - (- 2 x) + 3 = 0

$- (x^{2}) - (- 2 x) + 3 = 0$

y = - 2 + x^{2}

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

Step 3.3.1.4

Rewrite

3

$3$ as

- 1 (- 3)

$- 1 (- 3)$ .

- (x^{2}) - (- 2 x) - 1 \cdot - 3 = 0

$- (x^{2}) - (- 2 x) - 1 \cdot - 3 = 0$

y = - 2 + x^{2}

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

Step 3.3.1.5

Factor

- 1

$- 1$ out of

- (x^{2}) - (- 2 x)

$- (x^{2}) - (- 2 x)$ .

- (x^{2} - 2 x) - 1 \cdot - 3 = 0

$- (x^{2} - 2 x) - 1 \cdot - 3 = 0$

y = - 2 + x^{2}

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

Step 3.3.1.6

Factor

- 1

$- 1$ out of

- (x^{2} - 2 x) - 1 (- 3)

$- (x^{2} - 2 x) - 1 (- 3)$ .

- (x^{2} - 2 x - 3) = 0

$- (x^{2} - 2 x - 3) = 0$

y = - 2 + x^{2}

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

- (x^{2} - 2 x - 3) = 0

$- (x^{2} - 2 x - 3) = 0$

y = - 2 + x^{2}

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

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor

x^{2} - 2 x - 3

$x^{2} - 2 x - 3$ using the AC method.

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Step 3.3.2.1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 3

$- 3$ and whose sum is

- 2

$- 2$ .

- 3, 1

$- 3, 1$

y = - 2 + x^{2}

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

Step 3.3.2.1.2

Write the factored form using these integers.

- ((x - 3) (x + 1)) = 0

$- ((x - 3) (x + 1)) = 0$

y = - 2 + x^{2}

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

- ((x - 3) (x + 1)) = 0

$- ((x - 3) (x + 1)) = 0$

y = - 2 + x^{2}

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

Step 3.3.2.2

Remove unnecessary parentheses.

- (x - 3) (x + 1) = 0

$- (x - 3) (x + 1) = 0$

y = - 2 + x^{2}

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

- (x - 3) (x + 1) = 0

$- (x - 3) (x + 1) = 0$

y = - 2 + x^{2}

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

- (x - 3) (x + 1) = 0

$- (x - 3) (x + 1) = 0$

y = - 2 + x^{2}

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

Step 3.4

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x - 3 = 0

$x - 3 = 0$

x + 1 = 0

$x + 1 = 0$

y = - 2 + x^{2}

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

Step 3.5

Set

x - 3

$x - 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.5.1

Set

x - 3

$x - 3$ equal to

0

$0$ .

x - 3 = 0

$x - 3 = 0$

y = - 2 + x^{2}

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

Step 3.5.2

Add

3

$3$ to both sides of the equation.

x = 3

$x = 3$

y = - 2 + x^{2}

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

x = 3

$x = 3$

y = - 2 + x^{2}

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

Step 3.6

Set

x + 1

$x + 1$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.6.1

Set

x + 1

$x + 1$ equal to

0

$0$ .

x + 1 = 0

$x + 1 = 0$

y = - 2 + x^{2}

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

Step 3.6.2

Subtract

1

$1$ from both sides of the equation.

x = - 1

$x = - 1$

y = - 2 + x^{2}

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

x = - 1

$x = - 1$

y = - 2 + x^{2}

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

Step 3.7

The final solution is all the values that make

- (x - 3) (x + 1) = 0

$- (x - 3) (x + 1) = 0$ true.

x = 3, - 1

$x = 3, - 1$

y = - 2 + x^{2}

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

x = 3, - 1

$x = 3, - 1$

y = - 2 + x^{2}

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

Step 4

Replace all occurrences of

x

$x$ with

3

$3$ in each equation.

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Step 4.1

Replace all occurrences of

x

$x$ in

y = - 2 + x^{2}

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

3

$3$ .

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

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

x = 3

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify

- 2 + {(3)}^{2}

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

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Step 4.2.1.1

Raise

3

$3$ to the power of

2

$2$ .

y = - 2 + 9

$y = - 2 + 9$

x = 3

$x = 3$

Step 4.2.1.2

Add

- 2

$- 2$ and

9

$9$ .

y = 7

$y = 7$

x = 3

$x = 3$

y = 7

$y = 7$

x = 3

$x = 3$

y = 7

$y = 7$

x = 3

$x = 3$

y = 7

$y = 7$

x = 3

$x = 3$

Step 5

Replace all occurrences of

x

$x$ with

- 1

$- 1$ in each equation.

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Step 5.1

Replace all occurrences of

x

$x$ in

y = - 2 + x^{2}

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

- 1

$- 1$ .

y = - 2 + {(- 1)}^{2}

$y = - 2 + {(- 1)}^{2}$

x = - 1

$x = - 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify

- 2 + {(- 1)}^{2}

$- 2 + {(- 1)}^{2}$ .

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Step 5.2.1.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

y = - 2 + 1

$y = - 2 + 1$

x = - 1

$x = - 1$

Step 5.2.1.2

Add

- 2

$- 2$ and

1

$1$ .

y = - 1

$y = - 1$

x = - 1

$x = - 1$

y = - 1

$y = - 1$

x = - 1

$x = - 1$

y = - 1

$y = - 1$

x = - 1

$x = - 1$

y = - 1

$y = - 1$

x = - 1

$x = - 1$

Step 6

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

(3, 7)

$(3, 7)$

(- 1, - 1)

$(- 1, - 1)$

Step 7

The result can be shown in multiple forms.

Point Form:

(3, 7), (- 1, - 1)

$(3, 7), (- 1, - 1)$

Equation Form:

x = 3, y = 7

$x = 3, y = 7$

x = - 1, y = - 1

$x = - 1, y = - 1$

Step 8

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