Algebra Examples

$3 (y - 4) - 2 (x - 3) = - 6$ $5 x^{2} + 2 y^{2} - 53 = 0$

Step 1

Solve for $y$ in $3 (y - 4) - 2 (x - 3) = - 6$ .

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Step 1.1

Simplify $3 (y - 4) - 2 (x - 3)$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$3 y + 3 \cdot - 4 - 2 (x - 3) = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.1.1.2

Multiply $3$ by $- 4$ .

$3 y - 12 - 2 (x - 3) = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.1.1.3

Apply the distributive property.

$3 y - 12 - 2 x - 2 \cdot - 3 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.1.1.4

Multiply $- 2$ by $- 3$ .

$3 y - 12 - 2 x + 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y - 12 - 2 x + 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.1.2

Add $- 12$ and $6$ .

$3 y - 2 x - 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y - 2 x - 6 = - 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.2

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

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Step 1.2.1

Add $2 x$ to both sides of the equation.

$3 y - 6 = - 6 + 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.2.2

Add $6$ to both sides of the equation.

$3 y = - 6 + 2 x + 6$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.2.3

Combine the opposite terms in $- 6 + 2 x + 6$ .

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Step 1.2.3.1

Add $- 6$ and $6$ .

$3 y = 2 x + 0$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.2.3.2

Add $2 x$ and $0$ .

$3 y = 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y = 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

$3 y = 2 x$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.3

Divide each term in $3 y = 2 x$ by $3$ and simplify.

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Step 1.3.1

Divide each term in $3 y = 2 x$ by $3$ .

$\frac{3 y}{3} = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + 2 y^{2} - 53 = 0$

Step 2

Replace all occurrences of $y$ with $\frac{2 x}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $5 x^{2} + 2 y^{2} - 53 = 0$ with $\frac{2 x}{3}$ .

$5 x^{2} + 2 {(\frac{2 x}{3})}^{2} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 x^{2} + 2 {(\frac{2 x}{3})}^{2} - 53$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.1.1.1

Apply the product rule to $\frac{2 x}{3}$ .

$5 x^{2} + 2 (\frac{{(2 x)}^{2}}{3^{2}}) - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.1.1.2

Apply the product rule to $2 x$ .

$5 x^{2} + 2 (\frac{2^{2} x^{2}}{3^{2}}) - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + 2 (\frac{2^{2} x^{2}}{3^{2}}) - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.1.2

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

$5 x^{2} + 2 (\frac{4 x^{2}}{3^{2}}) - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.1.3

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

$5 x^{2} + 2 (\frac{4 x^{2}}{9}) - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.1.4

Multiply $2 \frac{4 x^{2}}{9}$ .

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Step 2.2.1.1.4.1

Combine $2$ and $\frac{4 x^{2}}{9}$ .

$5 x^{2} + \frac{2 (4 x^{2})}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.1.4.2

Multiply $4$ by $2$ .

$5 x^{2} + \frac{8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + \frac{8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$5 x^{2} + \frac{8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.2

To write $5 x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$5 x^{2} \cdot \frac{9}{9} + \frac{8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $5 x^{2}$ and $\frac{9}{9}$ .

$\frac{5 x^{2} \cdot 9}{9} + \frac{8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{5 x^{2} \cdot 9 + 8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{5 x^{2} \cdot 9 + 8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $x^{2}$ out of $5 x^{2} \cdot 9 + 8 x^{2}$ .

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Step 2.2.1.4.1.1.1

Factor $x^{2}$ out of $5 x^{2} \cdot 9$ .

$\frac{x^{2} (5 \cdot 9) + 8 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.4.1.1.2

Factor $x^{2}$ out of $8 x^{2}$ .

$\frac{x^{2} (5 \cdot 9) + x^{2} \cdot 8}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.4.1.1.3

Factor $x^{2}$ out of $x^{2} (5 \cdot 9) + x^{2} \cdot 8$ .

$\frac{x^{2} (5 \cdot 9 + 8)}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{x^{2} (5 \cdot 9 + 8)}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.4.1.2

Multiply $5$ by $9$ .

$\frac{x^{2} (45 + 8)}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.4.1.3

Add $45$ and $8$ .

$\frac{x^{2} \cdot 53}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{x^{2} \cdot 53}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 2.2.1.4.2

Move $53$ to the left of $x^{2}$ .

$\frac{53 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{53 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{53 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{53 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

$\frac{53 x^{2}}{9} - 53 = 0$

$y = \frac{2 x}{3}$

Step 3

Solve for $x$ in $\frac{53 x^{2}}{9} - 53 = 0$ .

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Step 3.1

Add $53$ to both sides of the equation.

$\frac{53 x^{2}}{9} = 53$

$y = \frac{2 x}{3}$

Step 3.2

Multiply both sides of the equation by $\frac{9}{53}$ .

$\frac{9}{53} \cdot \frac{53 x^{2}}{9} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{9}{53} \cdot \frac{53 x^{2}}{9}$ .

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Step 3.3.1.1.1

Combine.

$\frac{9 (53 x^{2})}{53 \cdot 9} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3.1.1.2

Cancel the common factor of $9$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{9 (53 x^{2})}{53 \cdot 9} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3.1.1.2.2

Rewrite the expression.

$\frac{53 x^{2}}{53} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

$\frac{53 x^{2}}{53} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3.1.1.3

Cancel the common factor of $53$ .

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Step 3.3.1.1.3.1

Cancel the common factor.

$\frac{53 x^{2}}{53} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3.1.1.3.2

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

$x^{2} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

$x^{2} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

$x^{2} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

$x^{2} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $53$ .

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Step 3.3.2.1.1

Cancel the common factor.

$x^{2} = \frac{9}{53} \cdot 53$

$y = \frac{2 x}{3}$

Step 3.3.2.1.2

Rewrite the expression.

$x^{2} = 9$

$y = \frac{2 x}{3}$

$x^{2} = 9$

$y = \frac{2 x}{3}$

$x^{2} = 9$

$y = \frac{2 x}{3}$

$x^{2} = 9$

$y = \frac{2 x}{3}$

Step 3.4

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

$x = \pm \sqrt{9}$

$y = \frac{2 x}{3}$

Step 3.5

Simplify $\pm \sqrt{9}$ .

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Step 3.5.1

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

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

$y = \frac{2 x}{3}$

Step 3.5.2

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

$x = \pm 3$

$y = \frac{2 x}{3}$

$x = \pm 3$

$y = \frac{2 x}{3}$

Step 3.6

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

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Step 3.6.1

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

$x = 3$

$y = \frac{2 x}{3}$

Step 3.6.2

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

$x = - 3$

$y = \frac{2 x}{3}$

Step 3.6.3

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

$x = 3, - 3$

$y = \frac{2 x}{3}$

$x = 3, - 3$

$y = \frac{2 x}{3}$

$x = 3, - 3$

$y = \frac{2 x}{3}$

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 $y = \frac{2 x}{3}$ with $3$ .

$y = \frac{2 (3)}{3}$

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$y = \frac{2 \cdot 3}{3}$

$x = 3$

Step 4.2.1.2

Divide $2$ by $1$ .

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

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 $y = \frac{2 x}{3}$ with $- 3$ .

$y = \frac{2 (- 3)}{3}$

$x = - 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{2 (- 3)}{3}$ .

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Step 5.2.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 5.2.1.1.1

Factor $3$ out of $2 (- 3)$ .

$y = \frac{3 (2 \cdot (- 1))}{3}$

$x = - 3$

Step 5.2.1.1.2

Cancel the common factors.

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Step 5.2.1.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 (2 \cdot (- 1))}{3 (1)}$

$x = - 3$

Step 5.2.1.1.2.2

Cancel the common factor.

$y = \frac{3 (2 \cdot (- 1))}{3 \cdot 1}$

$x = - 3$

Step 5.2.1.1.2.3

Rewrite the expression.

$y = \frac{2 \cdot (- 1)}{1}$

$x = - 3$

Step 5.2.1.1.2.4

Divide $2 \cdot (- 1)$ by $1$ .

$y = 2 \cdot (- 1)$

$x = - 3$

$y = 2 \cdot (- 1)$

$x = - 3$

$y = 2 \cdot (- 1)$

$x = - 3$

Step 5.2.1.2

Multiply $2$ by $- 1$ .

$y = - 2$

$x = - 3$

$y = - 2$

$x = - 3$

$y = - 2$

$x = - 3$

$y = - 2$

$x = - 3$

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