Algebra Examples

$3 x - 2 y = - 7$ $4 y = 9 - 6 x$

Step 1

Divide each term in $4 y = 9 - 6 x$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 y = 9 - 6 x$ by $4$ .

$\frac{4 y}{4} = \frac{9}{4} + \frac{- 6 x}{4}$

$3 x - 2 y = - 7$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{9}{4} + \frac{- 6 x}{4}$

$3 x - 2 y = - 7$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{9}{4} + \frac{- 6 x}{4}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} + \frac{- 6 x}{4}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} + \frac{- 6 x}{4}$

$3 x - 2 y = - 7$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $- 6$ and $4$ .

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Step 1.3.1.1.1

Factor $2$ out of $- 6 x$ .

$y = \frac{9}{4} + \frac{2 (- 3 x)}{4}$

$3 x - 2 y = - 7$

Step 1.3.1.1.2

Cancel the common factors.

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Step 1.3.1.1.2.1

Factor $2$ out of $4$ .

$y = \frac{9}{4} + \frac{2 (- 3 x)}{2 (2)}$

$3 x - 2 y = - 7$

Step 1.3.1.1.2.2

Cancel the common factor.

$y = \frac{9}{4} + \frac{2 (- 3 x)}{2 \cdot 2}$

$3 x - 2 y = - 7$

Step 1.3.1.1.2.3

Rewrite the expression.

$y = \frac{9}{4} + \frac{- 3 x}{2}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} + \frac{- 3 x}{2}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} + \frac{- 3 x}{2}$

$3 x - 2 y = - 7$

Step 1.3.1.2

Move the negative in front of the fraction.

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - 2 y = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - 2 y = - 7$

Step 2

Replace all occurrences of $y$ with $\frac{9}{4} - \frac{3 x}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $3 x - 2 y = - 7$ with $\frac{9}{4} - \frac{3 x}{2}$ .

$3 x - 2 (\frac{9}{4} - \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 x - 2 (\frac{9}{4} - \frac{3 x}{2})$ .

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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.

$3 x - 2 (\frac{9}{4}) - 2 (- \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $- 2$ .

$3 x + 2 (- 1) (\frac{9}{4}) - 2 (- \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.2.2

Factor $2$ out of $4$ .

$3 x + 2 \cdot (- 1 \frac{9}{2 \cdot 2}) - 2 (- \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.2.3

Cancel the common factor.

$3 x + 2 \cdot (- 1 \frac{9}{2 \cdot 2}) - 2 (- \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.2.4

Rewrite the expression.

$3 x - 1 (\frac{9}{2}) - 2 (- \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - 1 (\frac{9}{2}) - 2 (- \frac{3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

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

$3 x - 1 (\frac{9}{2}) - 2 \frac{- 3 x}{2} = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.3.2

Factor $2$ out of $- 2$ .

$3 x - 1 (\frac{9}{2}) + 2 (- 1) (\frac{- 3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.3.3

Cancel the common factor.

$3 x - 1 (\frac{9}{2}) + 2 \cdot (- 1 \frac{- 3 x}{2}) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.3.4

Rewrite the expression.

$3 x - 1 (\frac{9}{2}) - 1 (- 3 x) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - 1 (\frac{9}{2}) - 1 (- 3 x) = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.4

Multiply $- 3$ by $- 1$ .

$3 x - 1 (\frac{9}{2}) + 3 x = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.1.5

Rewrite $- 1 (\frac{9}{2})$ as $- (\frac{9}{2})$ .

$3 x - \frac{9}{2} + 3 x = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$3 x - \frac{9}{2} + 3 x = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 2.2.1.2

Add $3 x$ and $3 x$ .

$6 x - \frac{9}{2} = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$6 x - \frac{9}{2} = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$6 x - \frac{9}{2} = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

$6 x - \frac{9}{2} = - 7$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3

Solve for $x$ in $6 x - \frac{9}{2} = - 7$ .

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Step 3.1

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

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Step 3.1.1

Add $\frac{9}{2}$ to both sides of the equation.

$6 x = - 7 + \frac{9}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.1.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 x = - 7 \cdot \frac{2}{2} + \frac{9}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.1.3

Combine $- 7$ and $\frac{2}{2}$ .

$6 x = \frac{- 7 \cdot 2}{2} + \frac{9}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.1.4

Combine the numerators over the common denominator.

$6 x = \frac{- 7 \cdot 2 + 9}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $- 7$ by $2$ .

$6 x = \frac{- 14 + 9}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.1.5.2

Add $- 14$ and $9$ .

$6 x = \frac{- 5}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$6 x = \frac{- 5}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.1.6

Move the negative in front of the fraction.

$6 x = - \frac{5}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$6 x = - \frac{5}{2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.2

Divide each term in $6 x = - \frac{5}{2}$ by $6$ and simplify.

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Step 3.2.1

Divide each term in $6 x = - \frac{5}{2}$ by $6$ .

$\frac{6 x}{6} = \frac{- \frac{5}{2}}{6}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- \frac{5}{2}}{6}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{5}{2}}{6}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$x = \frac{- \frac{5}{2}}{6}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$x = \frac{- \frac{5}{2}}{6}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{5}{2} \cdot \frac{1}{6}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.2.3.2

Multiply $- \frac{5}{2} \cdot \frac{1}{6}$ .

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Step 3.2.3.2.1

Multiply $\frac{1}{6}$ by $\frac{5}{2}$ .

$x = - \frac{5}{6 \cdot 2}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 3.2.3.2.2

Multiply $6$ by $2$ .

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{3 x}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{3 x}{2}$

Step 4

Replace all occurrences of $x$ with $- \frac{5}{12}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{9}{4} - \frac{3 x}{2}$ with $- \frac{5}{12}$ .

$y = \frac{9}{4} - \frac{3 (- \frac{5}{12})}{2}$

$x = - \frac{5}{12}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{9}{4} - \frac{3 (- \frac{5}{12})}{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Simplify the numerator.

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Step 4.2.1.1.1.1

Multiply $- 1$ by $3$ .

$y = \frac{9}{4} - \frac{- 3 (\frac{5}{12})}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.1.2

Combine $- 3$ and $\frac{5}{12}$ .

$y = \frac{9}{4} - \frac{\frac{- 3 \cdot 5}{12}}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{\frac{- 3 \cdot 5}{12}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.2

Multiply $- 3$ by $5$ .

$y = \frac{9}{4} - \frac{\frac{- 15}{12}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.3

Reduce the expression by cancelling the common factors.

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Step 4.2.1.1.3.1

Cancel the common factor of $- 15$ and $12$ .

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Step 4.2.1.1.3.1.1

Factor $3$ out of $- 15$ .

$y = \frac{9}{4} - \frac{\frac{3 (- 5)}{12}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.3.1.2

Cancel the common factors.

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Step 4.2.1.1.3.1.2.1

Factor $3$ out of $12$ .

$y = \frac{9}{4} - \frac{\frac{3 \cdot - 5}{3 \cdot 4}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.3.1.2.2

Cancel the common factor.

$y = \frac{9}{4} - \frac{\frac{3 \cdot - 5}{3 \cdot 4}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.3.1.2.3

Rewrite the expression.

$y = \frac{9}{4} - \frac{\frac{- 5}{4}}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{\frac{- 5}{4}}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{\frac{- 5}{4}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.3.2

Move the negative in front of the fraction.

$y = \frac{9}{4} - \frac{- \frac{5}{4}}{2}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} - \frac{- \frac{5}{4}}{2}$

$x = - \frac{5}{12}$

Step 4.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{9}{4} - (- \frac{5}{4} \cdot \frac{1}{2})$

$x = - \frac{5}{12}$

Step 4.2.1.1.5

Multiply $- \frac{5}{4} \cdot \frac{1}{2}$ .

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Step 4.2.1.1.5.1

Multiply $\frac{1}{2}$ by $\frac{5}{4}$ .

$y = \frac{9}{4} + \frac{5}{2 \cdot 4}$

$x = - \frac{5}{12}$

Step 4.2.1.1.5.2

Multiply $2$ by $4$ .

$y = \frac{9}{4} + \frac{5}{8}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} + \frac{5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.1.6

Multiply $- - \frac{5}{8}$ .

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Step 4.2.1.1.6.1

Multiply $- 1$ by $- 1$ .

$y = \frac{9}{4} + 1 (\frac{5}{8})$

$x = - \frac{5}{12}$

Step 4.2.1.1.6.2

Multiply $\frac{5}{8}$ by $1$ .

$y = \frac{9}{4} + \frac{5}{8}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} + \frac{5}{8}$

$x = - \frac{5}{12}$

$y = \frac{9}{4} + \frac{5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.2

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{9}{4} \cdot \frac{2}{2} + \frac{5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{9}{4}$ by $\frac{2}{2}$ .

$y = \frac{9 \cdot 2}{4 \cdot 2} + \frac{5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.3.2

Multiply $4$ by $2$ .

$y = \frac{9 \cdot 2}{8} + \frac{5}{8}$

$x = - \frac{5}{12}$

$y = \frac{9 \cdot 2}{8} + \frac{5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{9 \cdot 2 + 5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $9$ by $2$ .

$y = \frac{18 + 5}{8}$

$x = - \frac{5}{12}$

Step 4.2.1.5.2

Add $18$ and $5$ .

$y = \frac{23}{8}$

$x = - \frac{5}{12}$

$y = \frac{23}{8}$

$x = - \frac{5}{12}$

$y = \frac{23}{8}$

$x = - \frac{5}{12}$

$y = \frac{23}{8}$

$x = - \frac{5}{12}$

$y = \frac{23}{8}$

$x = - \frac{5}{12}$

Step 5

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

$(- \frac{5}{12}, \frac{23}{8})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{5}{12}, \frac{23}{8})$

Equation Form:

$x = - \frac{5}{12}, y = \frac{23}{8}$

Step 7