Finite Math Examples

$- 5 x + 3 y = 6$ , $- 15 x + 9 y = 0$

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

Add $5 x$ to both sides of the equation.

$3 y = 6 + 5 x$

Step 1.3

Divide each term in $3 y = 6 + 5 x$ by $3$ and simplify.

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Step 1.3.1

Divide each term in $3 y = 6 + 5 x$ by $3$ .

$\frac{3 y}{3} = \frac{6}{3} + \frac{5 x}{3}$

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.

$\frac{3 y}{3} = \frac{6}{3} + \frac{5 x}{3}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{3} + \frac{5 x}{3}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $6$ by $3$ .

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

Step 1.4

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

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Step 1.4.1

Reorder $2$ and $\frac{5 x}{3}$ .

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

Step 1.4.2

Reorder terms.

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

Step 2

Using the slope-intercept form, the slope is $\frac{5}{3}$ .

$m_{1} = \frac{5}{3}$

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

Add $15 x$ to both sides of the equation.

$9 y = 15 x$

Step 3.3

Divide each term in $9 y = 15 x$ by $9$ and simplify.

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Step 3.3.1

Divide each term in $9 y = 15 x$ by $9$ .

$\frac{9 y}{9} = \frac{15 x}{9}$

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.

$\frac{9 y}{9} = \frac{15 x}{9}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{15 x}{9}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $15$ and $9$ .

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Step 3.3.3.1.1

Factor $3$ out of $15 x$ .

$y = \frac{3 (5 x)}{9}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $3$ out of $9$ .

$y = \frac{3 (5 x)}{3 (3)}$

Step 3.3.3.1.2.2

Cancel the common factor.

$y = \frac{3 (5 x)}{3 \cdot 3}$

Step 3.3.3.1.2.3

Rewrite the expression.

$y = \frac{5 x}{3}$

Step 3.4

Reorder terms.

$y = \frac{5}{3} x$

Step 4

Using the slope-intercept form, the slope is $\frac{5}{3}$ .

$m_{2} = \frac{5}{3}$

Step 5

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

$m_{1} = \frac{5}{3}$

$m_{2} = \frac{5}{3}$

Step 6