Finite Math Examples

$x - 3 y = 9$ , $3 x - y = 7$

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 $x$ from both sides of the equation.

$- 3 y = 9 - x$

Step 1.3

Divide each term in $- 3 y = 9 - x$ by $- 3$ and simplify.

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Step 1.3.1

Divide each term in $- 3 y = 9 - x$ by $- 3$ .

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

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.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

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 $9$ by $- 3$ .

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

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Reorder terms.

$y = \frac{1}{3} x - 3$

Step 2

Using the slope-intercept form, the slope is $\frac{1}{3}$ .

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

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 $3 x$ from both sides of the equation.

$- y = 7 - 3 x$

Step 3.3

Divide each term in $- y = 7 - 3 x$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- y = 7 - 3 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{7}{- 1} + \frac{- 3 x}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{7}{- 1} + \frac{- 3 x}{- 1}$

Step 3.3.2.2

Divide $y$ by $1$ .

$y = \frac{7}{- 1} + \frac{- 3 x}{- 1}$

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 $7$ by $- 1$ .

$y = - 7 + \frac{- 3 x}{- 1}$

Step 3.3.3.1.2

Move the negative one from the denominator of $\frac{- 3 x}{- 1}$ .

$y = - 7 - 1 \cdot (- 3 x)$

Step 3.3.3.1.3

Rewrite $- 1 \cdot (- 3 x)$ as $- (- 3 x)$ .

$y = - 7 - (- 3 x)$

Step 3.3.3.1.4

Multiply $- 3$ by $- 1$ .

$y = - 7 + 3 x$

Step 3.4

Reorder $- 7$ and $3 x$ .

$y = 3 x - 7$

Step 4

Using the slope-intercept form, the slope is $3$ .

$m_{2} = 3$

Step 5

Set up the system of equations to find any points of intersection.

$x - 3 y = 9, 3 x - y = 7$

Step 6

Solve the system of equations to find the point of intersection.

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Step 6.1

Add $3 y$ to both sides of the equation.

$x = 9 + 3 y$

$3 x - y = 7$

Step 6.2

Replace all occurrences of $x$ with $9 + 3 y$ in each equation.

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Step 6.2.1

Replace all occurrences of $x$ in $3 x - y = 7$ with $9 + 3 y$ .

$3 (9 + 3 y) - y = 7$

$x = 9 + 3 y$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify $3 (9 + 3 y) - y$ .

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Step 6.2.2.1.1

Simplify each term.

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Step 6.2.2.1.1.1

Apply the distributive property.

$3 \cdot 9 + 3 (3 y) - y = 7$

$x = 9 + 3 y$

Step 6.2.2.1.1.2

Multiply $3$ by $9$ .

$27 + 3 (3 y) - y = 7$

$x = 9 + 3 y$

Step 6.2.2.1.1.3

Multiply $3$ by $3$ .

$27 + 9 y - y = 7$

$x = 9 + 3 y$

$27 + 9 y - y = 7$

$x = 9 + 3 y$

Step 6.2.2.1.2

Subtract $y$ from $9 y$ .

$27 + 8 y = 7$

$x = 9 + 3 y$

$27 + 8 y = 7$

$x = 9 + 3 y$

$27 + 8 y = 7$

$x = 9 + 3 y$

$27 + 8 y = 7$

$x = 9 + 3 y$

Step 6.3

Solve for $y$ in $27 + 8 y = 7$ .

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Step 6.3.1

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

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Step 6.3.1.1

Subtract $27$ from both sides of the equation.

$8 y = 7 - 27$

$x = 9 + 3 y$

Step 6.3.1.2

Subtract $27$ from $7$ .

$8 y = - 20$

$x = 9 + 3 y$

$8 y = - 20$

$x = 9 + 3 y$

Step 6.3.2

Divide each term in $8 y = - 20$ by $8$ and simplify.

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Step 6.3.2.1

Divide each term in $8 y = - 20$ by $8$ .

$\frac{8 y}{8} = \frac{- 20}{8}$

$x = 9 + 3 y$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $8$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 20}{8}$

$x = 9 + 3 y$

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 20}{8}$

$x = 9 + 3 y$

$y = \frac{- 20}{8}$

$x = 9 + 3 y$

$y = \frac{- 20}{8}$

$x = 9 + 3 y$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Cancel the common factor of $- 20$ and $8$ .

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Step 6.3.2.3.1.1

Factor $4$ out of $- 20$ .

$y = \frac{4 (- 5)}{8}$

$x = 9 + 3 y$

Step 6.3.2.3.1.2

Cancel the common factors.

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Step 6.3.2.3.1.2.1

Factor $4$ out of $8$ .

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

$x = 9 + 3 y$

Step 6.3.2.3.1.2.2

Cancel the common factor.

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

$x = 9 + 3 y$

Step 6.3.2.3.1.2.3

Rewrite the expression.

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

$x = 9 + 3 y$

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

$x = 9 + 3 y$

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

$x = 9 + 3 y$

Step 6.3.2.3.2

Move the negative in front of the fraction.

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

$x = 9 + 3 y$

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

$x = 9 + 3 y$

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

$x = 9 + 3 y$

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

$x = 9 + 3 y$

Step 6.4

Replace all occurrences of $y$ with $- \frac{5}{2}$ in each equation.

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Step 6.4.1

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

$x = 9 + 3 (- \frac{5}{2})$

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

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

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

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Step 6.4.2.1.1

Simplify each term.

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Step 6.4.2.1.1.1

Multiply $3 (- \frac{5}{2})$ .

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Step 6.4.2.1.1.1.1

Multiply $- 1$ by $3$ .

$x = 9 - 3 (\frac{5}{2})$

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

Step 6.4.2.1.1.1.2

Combine $- 3$ and $\frac{5}{2}$ .

$x = 9 + \frac{- 3 \cdot 5}{2}$

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

Step 6.4.2.1.1.1.3

Multiply $- 3$ by $5$ .

$x = 9 + \frac{- 15}{2}$

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

$x = 9 + \frac{- 15}{2}$

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

Step 6.4.2.1.1.2

Move the negative in front of the fraction.

$x = 9 - \frac{15}{2}$

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

$x = 9 - \frac{15}{2}$

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

Step 6.4.2.1.2

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

$x = 9 \cdot \frac{2}{2} - \frac{15}{2}$

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

Step 6.4.2.1.3

Combine $9$ and $\frac{2}{2}$ .

$x = \frac{9 \cdot 2}{2} - \frac{15}{2}$

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

Step 6.4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{9 \cdot 2 - 15}{2}$

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

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Multiply $9$ by $2$ .

$x = \frac{18 - 15}{2}$

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

Step 6.4.2.1.5.2

Subtract $15$ from $18$ .

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

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

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

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

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

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

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

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

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

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

Step 6.5

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

$(\frac{3}{2}, - \frac{5}{2})$

Step 7

Since the slopes are different, the lines will have exactly one intersection point.

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

$m_{2} = 3$

$(\frac{3}{2}, - \frac{5}{2})$

Step 8