Algebra Examples

$4 x^{2} - 3 y^{2} + 19 y - 44 = 0$ $x + y = 2$

Step 1

Subtract $y$ from both sides of the equation.

$x = 2 - y$

$4 x^{2} - 3 y^{2} + 19 y - 44 = 0$

Step 2

Replace all occurrences of $x$ with $2 - y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x^{2} - 3 y^{2} + 19 y - 44 = 0$ with $2 - y$ .

$4 {(2 - y)}^{2} - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 {(2 - y)}^{2} - 3 y^{2} + 19 y - 44$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(2 - y)}^{2}$ as $(2 - y) (2 - y)$ .

$4 ((2 - y) (2 - y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.2

Expand $(2 - y) (2 - y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$4 (2 (2 - y) - y (2 - y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$4 (2 \cdot 2 + 2 (- y) - y (2 - y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$4 (2 \cdot 2 + 2 (- y) - y \cdot 2 - y (- y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

$4 (2 \cdot 2 + 2 (- y) - y \cdot 2 - y (- y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $2$ by $2$ .

$4 (4 + 2 (- y) - y \cdot 2 - y (- y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $2$ .

$4 (4 - 2 y - y \cdot 2 - y (- y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.3

Multiply $2$ by $- 1$ .

$4 (4 - 2 y - 2 y - y (- y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 (4 - 2 y - 2 y - 1 \cdot (- 1 y \cdot y)) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.1.3.1.5.1

Move $y$ .

$4 (4 - 2 y - 2 y - 1 \cdot (- 1 (y \cdot y))) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$4 (4 - 2 y - 2 y - 1 \cdot (- 1 y^{2})) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

$4 (4 - 2 y - 2 y - 1 \cdot (- 1 y^{2})) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 (4 - 2 y - 2 y + 1 y^{2}) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.1.7

Multiply $y^{2}$ by $1$ .

$4 (4 - 2 y - 2 y + y^{2}) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

$4 (4 - 2 y - 2 y + y^{2}) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.3.2

Subtract $2 y$ from $- 2 y$ .

$4 (4 - 4 y + y^{2}) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

$4 (4 - 4 y + y^{2}) - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.4

Apply the distributive property.

$4 \cdot 4 + 4 (- 4 y) + 4 y^{2} - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.5

Simplify.

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Step 2.2.1.1.5.1

Multiply $4$ by $4$ .

$16 + 4 (- 4 y) + 4 y^{2} - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.1.5.2

Multiply $- 4$ by $4$ .

$16 - 16 y + 4 y^{2} - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

$16 - 16 y + 4 y^{2} - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

$16 - 16 y + 4 y^{2} - 3 y^{2} + 19 y - 44 = 0$

$x = 2 - y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $44$ from $16$ .

$- 16 y + 4 y^{2} - 3 y^{2} + 19 y - 28 = 0$

$x = 2 - y$

Step 2.2.1.2.2

Add $- 16 y$ and $19 y$ .

$4 y^{2} - 3 y^{2} + 3 y - 28 = 0$

$x = 2 - y$

Step 2.2.1.2.3

Subtract $3 y^{2}$ from $4 y^{2}$ .

$y^{2} + 3 y - 28 = 0$

$x = 2 - y$

$y^{2} + 3 y - 28 = 0$

$x = 2 - y$

$y^{2} + 3 y - 28 = 0$

$x = 2 - y$

$y^{2} + 3 y - 28 = 0$

$x = 2 - y$

$y^{2} + 3 y - 28 = 0$

$x = 2 - y$

Step 3

Solve for $y$ in $y^{2} + 3 y - 28 = 0$ .

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Step 3.1

Factor $y^{2} + 3 y - 28$ using the AC method.

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Step 3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 28$ and whose sum is $3$ .

$- 4, 7$

$x = 2 - y$

Step 3.1.2

Write the factored form using these integers.

$(y - 4) (y + 7) = 0$

$x = 2 - y$

$(y - 4) (y + 7) = 0$

$x = 2 - y$

Step 3.2

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

$y - 4 = 0$

$y + 7 = 0$

$x = 2 - y$

Step 3.3

Set $y - 4$ equal to $0$ and solve for $y$ .

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Step 3.3.1

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

$x = 2 - y$

Step 3.3.2

Add $4$ to both sides of the equation.

$y = 4$

$x = 2 - y$

$y = 4$

$x = 2 - y$

Step 3.4

Set $y + 7$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $y + 7$ equal to $0$ .

$y + 7 = 0$

$x = 2 - y$

Step 3.4.2

Subtract $7$ from both sides of the equation.

$y = - 7$

$x = 2 - y$

$y = - 7$

$x = 2 - y$

Step 3.5

The final solution is all the values that make $(y - 4) (y + 7) = 0$ true.

$y = 4, - 7$

$x = 2 - y$

$y = 4, - 7$

$x = 2 - y$

Step 4

Replace all occurrences of $y$ with $4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 - y$ with $4$ .

$x = 2 - (4)$

$y = 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - (4)$ .

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Step 4.2.1.1

Multiply $- 1$ by $4$ .

$x = 2 - 4$

$y = 4$

Step 4.2.1.2

Subtract $4$ from $2$ .

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

Step 5

Replace all occurrences of $y$ with $- 7$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 2 - y$ with $- 7$ .

$x = 2 - (- 7)$

$y = - 7$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 - (- 7)$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 7$ .

$x = 2 + 7$

$y = - 7$

Step 5.2.1.2

Add $2$ and $7$ .

$x = 9$

$y = - 7$

$x = 9$

$y = - 7$

$x = 9$

$y = - 7$

$x = 9$

$y = - 7$

Step 6

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

$(- 2, 4)$

$(9, - 7)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 2, 4), (9, - 7)$

Equation Form:

$x = - 2, y = 4$

$x = 9, y = - 7$

Step 8