Finite Math Examples

$2 x + 3 y = - 8$ , $6 x + 9 y = 2$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

Subtract $2 x$ from both sides of the equation.

$3 y = - 8 - 2 x$

Step 1.3

Divide each term in $3 y = - 8 - 2 x$ by $3$ and simplify.

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Step 1.3.1

Divide each term in $3 y = - 8 - 2 x$ by $3$ .

$\frac{3 y}{3} = \frac{- 8}{3} + \frac{- 2 x}{3}$

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.

$\frac{3 y}{3} = \frac{- 8}{3} + \frac{- 2 x}{3}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 8}{3} + \frac{- 2 x}{3}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

$y = - \frac{8}{3} + \frac{- 2 x}{3}$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{8}{3} - \frac{2 x}{3}$

Step 1.4

Write in $y = m x + b$ form.

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Step 1.4.1

Reorder $- \frac{8}{3}$ and $- \frac{2 x}{3}$ .

$y = - \frac{2 x}{3} - \frac{8}{3}$

Step 1.4.2

Reorder terms.

$y = - (\frac{2}{3} x) - \frac{8}{3}$

Step 1.4.3

Remove parentheses.

$y = - \frac{2}{3} x - \frac{8}{3}$

Step 2

Using the slope-intercept form, the slope is $- \frac{2}{3}$ .

$m_{1} = - \frac{2}{3}$

Step 3

Rewrite in slope-intercept form.

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Step 3.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 3.2

Subtract $6 x$ from both sides of the equation.

$9 y = 2 - 6 x$

Step 3.3

Divide each term in $9 y = 2 - 6 x$ by $9$ and simplify.

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Step 3.3.1

Divide each term in $9 y = 2 - 6 x$ by $9$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $- 6$ and $9$ .

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Step 3.3.3.1.1.1

Factor $3$ out of $- 6 x$ .

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

Step 3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.1.1.2.1

Factor $3$ out of $9$ .

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

Step 3.3.3.1.1.2.2

Cancel the common factor.

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

Step 3.3.3.1.1.2.3

Rewrite the expression.

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

Step 3.3.3.1.2

Move the negative in front of the fraction.

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

Step 3.4

Write in $y = m x + b$ form.

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Step 3.4.1

Reorder $\frac{2}{9}$ and $- \frac{2 x}{3}$ .

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

Step 3.4.2

Reorder terms.

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

Step 3.4.3

Remove parentheses.

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

Step 4

Using the slope-intercept form, the slope is $- \frac{2}{3}$ .

$m_{2} = - \frac{2}{3}$

Step 5

Since the slopes are the same, the lines are parallel and do not intersect, so the system has no solutions.

$m_{1} = - \frac{2}{3}$

$m_{2} = - \frac{2}{3}$

Step 6