Finite Math Examples

$2 x - 3 y = 12$ , $4 x - 6 y = - 24$

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.

$- 3 y = 12 - 2 x$

Step 1.3

Divide each term in $- 3 y = 12 - 2 x$ by $- 3$ and simplify.

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Step 1.3.1

Divide each term in $- 3 y = 12 - 2 x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{12}{- 3} + \frac{- 2 x}{- 3}$

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.

$\frac{- 3 y}{- 3} = \frac{12}{- 3} + \frac{- 2 x}{- 3}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{- 3} + \frac{- 2 x}{- 3}$

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

Divide $12$ by $- 3$ .

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

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.4

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

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Step 1.4.1

Reorder $- 4$ and $\frac{2 x}{3}$ .

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

Step 1.4.2

Reorder terms.

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

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 $4 x$ from both sides of the equation.

$- 6 y = - 24 - 4 x$

Step 3.3

Divide each term in $- 6 y = - 24 - 4 x$ by $- 6$ and simplify.

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Step 3.3.1

Divide each term in $- 6 y = - 24 - 4 x$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{- 24}{- 6} + \frac{- 4 x}{- 6}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 6$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{- 24}{- 6} + \frac{- 4 x}{- 6}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 24}{- 6} + \frac{- 4 x}{- 6}$

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

Divide $- 24$ by $- 6$ .

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

Step 3.3.3.1.2

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

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Step 3.3.3.1.2.1

Factor $- 2$ out of $- 4 x$ .

$y = 4 + \frac{- 2 (2 x)}{- 6}$

Step 3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.1.2.2.1

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

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

Step 3.3.3.1.2.2.2

Cancel the common factor.

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

Step 3.3.3.1.2.2.3

Rewrite the expression.

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

Step 3.4

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

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Step 3.4.1

Reorder $4$ and $\frac{2 x}{3}$ .

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

Step 3.4.2

Reorder terms.

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

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