Algebra Examples

$3 y + x^{2} - 4 x - 17 = 0$ $3 y - 10 x + 38 = 0$

Step 1

Solve for $y$ in $3 y + x^{2} - 4 x - 17 = 0$ .

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Step 1.1

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

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Step 1.1.1

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

$3 y - 4 x - 17 = - x^{2}$

$3 y - 10 x + 38 = 0$

Step 1.1.2

Add $4 x$ to both sides of the equation.

$3 y - 17 = - x^{2} + 4 x$

$3 y - 10 x + 38 = 0$

Step 1.1.3

Add $17$ to both sides of the equation.

$3 y = - x^{2} + 4 x + 17$

$3 y - 10 x + 38 = 0$

$3 y = - x^{2} + 4 x + 17$

$3 y - 10 x + 38 = 0$

Step 1.2

Divide each term in $3 y = - x^{2} + 4 x + 17$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 y = - x^{2} + 4 x + 17$ by $3$ .

$\frac{3 y}{3} = \frac{- x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

$y = \frac{- x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

$y = \frac{- x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$3 y - 10 x + 38 = 0$

Step 2

Replace all occurrences of $y$ with $- \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $3 y - 10 x + 38 = 0$ with $- \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$ .

$3 (- \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}) - 10 x + 38$ .

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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.

$3 (- \frac{x^{2}}{3}) + 3 (\frac{4 x}{3}) + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1.1

Move the leading negative in $- \frac{x^{2}}{3}$ into the numerator.

$3 (\frac{- x^{2}}{3}) + 3 (\frac{4 x}{3}) + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2.1.2

Cancel the common factor.

$3 (\frac{- x^{2}}{3}) + 3 (\frac{4 x}{3}) + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2.1.3

Rewrite the expression.

$- x^{2} + 3 (\frac{4 x}{3}) + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} + 3 (\frac{4 x}{3}) + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.2.1

Cancel the common factor.

$- x^{2} + 3 (\frac{4 x}{3}) + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2.2.2

Rewrite the expression.

$- x^{2} + 4 x + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} + 4 x + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.3.1

Cancel the common factor.

$- x^{2} + 4 x + 3 (\frac{17}{3}) - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.1.2.3.2

Rewrite the expression.

$- x^{2} + 4 x + 17 - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} + 4 x + 17 - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} + 4 x + 17 - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} + 4 x + 17 - 10 x + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $10 x$ from $4 x$ .

$- x^{2} - 6 x + 17 + 38 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 2.2.1.2.2

Add $17$ and $38$ .

$- x^{2} - 6 x + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} - 6 x + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} - 6 x + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} - 6 x + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- x^{2} - 6 x + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3

Solve for $x$ in $- x^{2} - 6 x + 55 = 0$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Factor $- 1$ out of $- x^{2} - 6 x + 55$ .

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Step 3.1.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 6 x + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.1.2

Factor $- 1$ out of $- 6 x$ .

$- (x^{2}) - (6 x) + 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.1.3

Rewrite $55$ as $- 1 (- 55)$ .

$- (x^{2}) - (6 x) - 1 \cdot - 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.1.4

Factor $- 1$ out of $- (x^{2}) - (6 x)$ .

$- (x^{2} + 6 x) - 1 \cdot - 55 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.1.5

Factor $- 1$ out of $- (x^{2} + 6 x) - 1 (- 55)$ .

$- (x^{2} + 6 x - 55) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- (x^{2} + 6 x - 55) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.2

Factor.

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Step 3.1.2.1

Factor $x^{2} + 6 x - 55$ using the AC method.

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Step 3.1.2.1.1

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 $- 55$ and whose sum is $6$ .

$- 5, 11$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.2.1.2

Write the factored form using these integers.

$- ((x - 5) (x + 11)) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- ((x - 5) (x + 11)) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.1.2.2

Remove unnecessary parentheses.

$- (x - 5) (x + 11) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- (x - 5) (x + 11) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$- (x - 5) (x + 11) = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

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 $0$ .

$x - 5 = 0$

$x + 11 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.3.2

Add $5$ to both sides of the equation.

$x = 5$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$x = 5$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.4

Set $x + 11$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.4.2

Subtract $11$ from both sides of the equation.

$x = - 11$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$x = - 11$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 3.5

The final solution is all the values that make $- (x - 5) (x + 11) = 0$ true.

$x = 5, - 11$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

$x = 5, - 11$

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$

Step 4

Replace all occurrences of $x$ with $5$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$ with $5$ .

$y = - \frac{{(5)}^{2}}{3} + \frac{4 (5)}{3} + \frac{17}{3}$

$x = 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{{(5)}^{2}}{3} + \frac{4 (5)}{3} + \frac{17}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- {(5)}^{2} + 4 (5) + 17}{3}$

$x = 5$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Raise $5$ to the power of $2$ .

$y = \frac{- 1 \cdot 25 + 4 (5) + 17}{3}$

$x = 5$

Step 4.2.1.2.2

Multiply $- 1$ by $25$ .

$y = \frac{- 25 + 4 (5) + 17}{3}$

$x = 5$

Step 4.2.1.2.3

Multiply $4$ by $5$ .

$y = \frac{- 25 + 20 + 17}{3}$

$x = 5$

$y = \frac{- 25 + 20 + 17}{3}$

$x = 5$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Add $- 25$ and $20$ .

$y = \frac{- 5 + 17}{3}$

$x = 5$

Step 4.2.1.3.2

Add $- 5$ and $17$ .

$y = \frac{12}{3}$

$x = 5$

Step 4.2.1.3.3

Divide $12$ by $3$ .

$y = 4$

$x = 5$

$y = 4$

$x = 5$

$y = 4$

$x = 5$

$y = 4$

$x = 5$

$y = 4$

$x = 5$

Step 5

Replace all occurrences of $x$ with $- 11$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{17}{3}$ with $- 11$ .

$y = - \frac{{(- 11)}^{2}}{3} + \frac{4 (- 11)}{3} + \frac{17}{3}$

$x = - 11$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{{(- 11)}^{2}}{3} + \frac{4 (- 11)}{3} + \frac{17}{3}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- {(- 11)}^{2} + 4 (- 11) + 17}{3}$

$x = - 11$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

Raise $- 11$ to the power of $2$ .

$y = \frac{- 1 \cdot 121 + 4 (- 11) + 17}{3}$

$x = - 11$

Step 5.2.1.2.2

Multiply $- 1$ by $121$ .

$y = \frac{- 121 + 4 (- 11) + 17}{3}$

$x = - 11$

Step 5.2.1.2.3

Multiply $4$ by $- 11$ .

$y = \frac{- 121 - 44 + 17}{3}$

$x = - 11$

$y = \frac{- 121 - 44 + 17}{3}$

$x = - 11$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Subtract $44$ from $- 121$ .

$y = \frac{- 165 + 17}{3}$

$x = - 11$

Step 5.2.1.3.2

Add $- 165$ and $17$ .

$y = \frac{- 148}{3}$

$x = - 11$

Step 5.2.1.3.3

Move the negative in front of the fraction.

$y = - \frac{148}{3}$

$x = - 11$

$y = - \frac{148}{3}$

$x = - 11$

$y = - \frac{148}{3}$

$x = - 11$

$y = - \frac{148}{3}$

$x = - 11$

$y = - \frac{148}{3}$

$x = - 11$

Step 6

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

$(5, 4)$

$(- 11, - \frac{148}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(5, 4), (- 11, - \frac{148}{3})$

Equation Form:

$x = 5, y = 4$

$x = - 11, y = - \frac{148}{3}$

Step 8