Algebra Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

$\frac{2}{3 x} - 2 = - x + 3$

Step 2

Solve $\frac{2}{3 x} - 2 = - x + 3$ for $x$ .

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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

Add $2$ to both sides of the equation.

$\frac{2}{3 x} = - x + 3 + 2$

Step 2.1.2

Add $3$ and $2$ .

$\frac{2}{3 x} = - x + 5$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 x, 1, 1 + y = - x + 3$

Step 2.2.2

The LCM of one and any expression is the expression.

$3 x + y = - x + 3$

Step 2.3

Multiply each term in $\frac{2}{3 x} = - x + 5$ by $3 x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{2}{3 x} = - x + 5$ by $3 x$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.3.2.2

Cancel the common factor of $3$ .

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Step 2.3.2.2.1

Factor $3$ out of $3 x$ .

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.3

Cancel the common factor of $x$ .

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Step 2.3.2.3.1

Cancel the common factor.

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

Step 2.3.2.3.2

Rewrite the expression.

$2 = - x (3 x) + 5 (3 x)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Rewrite using the commutative property of multiplication.

$2 = - 1 \cdot 3 x \cdot x + 5 (3 x)$

Step 2.3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.3.1.2.1

Move $x$ .

$2 = - 1 \cdot 3 (x \cdot x) + 5 (3 x)$

Step 2.3.3.1.2.2

Multiply $x$ by $x$ .

$2 = - 1 \cdot 3 x^{2} + 5 (3 x)$

Step 2.3.3.1.3

Multiply $- 1$ by $3$ .

$2 = - 3 x^{2} + 5 (3 x)$

Step 2.3.3.1.4

Multiply $3$ by $5$ .

$2 = - 3 x^{2} + 15 x$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $- 3 x^{2} + 15 x = 2$ .

$- 3 x^{2} + 15 x = 2$

Step 2.4.2

Subtract $2$ from both sides of the equation.

$- 3 x^{2} + 15 x - 2 = 0$

Step 2.4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = - x + 3$

Step 2.4.4

Substitute the values $a = - 3$ , $b = 15$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 15 \pm \sqrt{15^{2} - 4 \cdot (- 3 \cdot - 2)}}{2 \cdot - 3} + y = - x + 3$

Step 2.4.5

Simplify.

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Step 2.4.5.1

Simplify the numerator.

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Step 2.4.5.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot - 3 \cdot - 2}}{2 \cdot - 3}$

Step 2.4.5.1.2

Multiply $- 4 \cdot - 3 \cdot - 2$ .

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Step 2.4.5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 15 \pm \sqrt{225 + 12 \cdot - 2}}{2 \cdot - 3}$

Step 2.4.5.1.2.2

Multiply $12$ by $- 2$ .

$x = \frac{- 15 \pm \sqrt{225 - 24}}{2 \cdot - 3}$

Step 2.4.5.1.3

Subtract $24$ from $225$ .

$x = \frac{- 15 \pm \sqrt{201}}{2 \cdot - 3}$

Step 2.4.5.2

Multiply $2$ by $- 3$ .

$x = \frac{- 15 \pm \sqrt{201}}{- 6}$

Step 2.4.5.3

Simplify $\frac{- 15 \pm \sqrt{201}}{- 6}$ .

$x = \frac{15 \pm \sqrt{201}}{6}$

Step 2.4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.6.1

Simplify the numerator.

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Step 2.4.6.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot - 3 \cdot - 2}}{2 \cdot - 3}$

Step 2.4.6.1.2

Multiply $- 4 \cdot - 3 \cdot - 2$ .

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Step 2.4.6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 15 \pm \sqrt{225 + 12 \cdot - 2}}{2 \cdot - 3}$

Step 2.4.6.1.2.2

Multiply $12$ by $- 2$ .

$x = \frac{- 15 \pm \sqrt{225 - 24}}{2 \cdot - 3}$

Step 2.4.6.1.3

Subtract $24$ from $225$ .

$x = \frac{- 15 \pm \sqrt{201}}{2 \cdot - 3}$

Step 2.4.6.2

Multiply $2$ by $- 3$ .

$x = \frac{- 15 \pm \sqrt{201}}{- 6}$

Step 2.4.6.3

Simplify $\frac{- 15 \pm \sqrt{201}}{- 6}$ .

$x = \frac{15 \pm \sqrt{201}}{6}$

Step 2.4.6.4

Change the $\pm$ to $+$ .

$x = \frac{15 + \sqrt{201}}{6}$

Step 2.4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.7.1

Simplify the numerator.

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Step 2.4.7.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot - 3 \cdot - 2}}{2 \cdot - 3}$

Step 2.4.7.1.2

Multiply $- 4 \cdot - 3 \cdot - 2$ .

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Step 2.4.7.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 15 \pm \sqrt{225 + 12 \cdot - 2}}{2 \cdot - 3}$

Step 2.4.7.1.2.2

Multiply $12$ by $- 2$ .

$x = \frac{- 15 \pm \sqrt{225 - 24}}{2 \cdot - 3}$

Step 2.4.7.1.3

Subtract $24$ from $225$ .

$x = \frac{- 15 \pm \sqrt{201}}{2 \cdot - 3}$

Step 2.4.7.2

Multiply $2$ by $- 3$ .

$x = \frac{- 15 \pm \sqrt{201}}{- 6}$

Step 2.4.7.3

Simplify $\frac{- 15 \pm \sqrt{201}}{- 6}$ .

$x = \frac{15 \pm \sqrt{201}}{6}$

Step 2.4.7.4

Change the $\pm$ to $-$ .

$x = \frac{15 - \sqrt{201}}{6}$

Step 2.4.8

The final answer is the combination of both solutions.

$x = \frac{15 + \sqrt{201}}{6}, \frac{15 - \sqrt{201}}{6}$

Step 3

Evaluate $y$ when $x = \frac{15 + \sqrt{201}}{6}$ .

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Step 3.1

Substitute $\frac{15 + \sqrt{201}}{6}$ for $x$ .

$y = - (\frac{15 + \sqrt{201}}{6}) + 3$

Step 3.2

Substitute $\frac{15 + \sqrt{201}}{6}$ for $x$ in $y = - (\frac{15 + \sqrt{201}}{6}) + 3$ and solve for $y$ .

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Step 3.2.1

Multiply $- 1$ by $\frac{15 + \sqrt{201}}{6}$ .

$y = - \frac{15 + \sqrt{201}}{6} + 3$

Step 3.2.2

Simplify $- \frac{15 + \sqrt{201}}{6} + 3$ .

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Step 3.2.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - \frac{15 + \sqrt{201}}{6} + 3 \cdot \frac{6}{6}$

Step 3.2.2.2

Combine fractions.

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Step 3.2.2.2.1

Combine $3$ and $\frac{6}{6}$ .

$y = - \frac{15 + \sqrt{201}}{6} + \frac{3 \cdot 6}{6}$

Step 3.2.2.2.2

Combine the numerators over the common denominator.

$y = \frac{- (15 + \sqrt{201}) + 3 \cdot 6}{6}$

Step 3.2.2.3

Simplify the numerator.

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Step 3.2.2.3.1

Apply the distributive property.

$y = \frac{- 1 \cdot 15 - \sqrt{201} + 3 \cdot 6}{6}$

Step 3.2.2.3.2

Multiply $- 1$ by $15$ .

$y = \frac{- 15 - \sqrt{201} + 3 \cdot 6}{6}$

Step 3.2.2.3.3

Multiply $3$ by $6$ .

$y = \frac{- 15 - \sqrt{201} + 18}{6}$

Step 3.2.2.3.4

Add $- 15$ and $18$ .

$y = \frac{3 - \sqrt{201}}{6}$

Step 4

Evaluate $y$ when $x = \frac{15 - \sqrt{201}}{6}$ .

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Step 4.1

Substitute $\frac{15 - \sqrt{201}}{6}$ for $x$ .

$y = - (\frac{15 - \sqrt{201}}{6}) + 3$

Step 4.2

Substitute $\frac{15 - \sqrt{201}}{6}$ for $x$ in $y = - (\frac{15 - \sqrt{201}}{6}) + 3$ and solve for $y$ .

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Step 4.2.1

Multiply $- 1$ by $\frac{15 - \sqrt{201}}{6}$ .

$y = - \frac{15 - \sqrt{201}}{6} + 3$

Step 4.2.2

Simplify $- \frac{15 - \sqrt{201}}{6} + 3$ .

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Step 4.2.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - \frac{15 - \sqrt{201}}{6} + 3 \cdot \frac{6}{6}$

Step 4.2.2.2

Combine fractions.

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Step 4.2.2.2.1

Combine $3$ and $\frac{6}{6}$ .

$y = - \frac{15 - \sqrt{201}}{6} + \frac{3 \cdot 6}{6}$

Step 4.2.2.2.2

Combine the numerators over the common denominator.

$y = \frac{- (15 - \sqrt{201}) + 3 \cdot 6}{6}$

Step 4.2.2.3

Simplify the numerator.

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Step 4.2.2.3.1

Apply the distributive property.

$y = \frac{- 1 \cdot 15 - - \sqrt{201} + 3 \cdot 6}{6}$

Step 4.2.2.3.2

Multiply $- 1$ by $15$ .

$y = \frac{- 15 - - \sqrt{201} + 3 \cdot 6}{6}$

Step 4.2.2.3.3

Multiply $- - \sqrt{201}$ .

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Step 4.2.2.3.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 15 + 1 \sqrt{201} + 3 \cdot 6}{6}$

Step 4.2.2.3.3.2

Multiply $\sqrt{201}$ by $1$ .

$y = \frac{- 15 + \sqrt{201} + 3 \cdot 6}{6}$

Step 4.2.2.3.4

Multiply $3$ by $6$ .

$y = \frac{- 15 + \sqrt{201} + 18}{6}$

Step 4.2.2.3.5

Add $- 15$ and $18$ .

$y = \frac{3 + \sqrt{201}}{6}$

Step 5

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

$(\frac{15 + \sqrt{201}}{6}, \frac{3 - \sqrt{201}}{6})$

$(\frac{15 - \sqrt{201}}{6}, \frac{3 + \sqrt{201}}{6})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{15 + \sqrt{201}}{6}, \frac{3 - \sqrt{201}}{6}), (\frac{15 - \sqrt{201}}{6}, \frac{3 + \sqrt{201}}{6})$

Equation Form:

$x = \frac{15 + \sqrt{201}}{6}, y = \frac{3 - \sqrt{201}}{6}$

$x = \frac{15 - \sqrt{201}}{6}, y = \frac{3 + \sqrt{201}}{6}$

Step 7