Finite Math Examples

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

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

Simplify the right side.

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Step 1.2.1

Combine $x$ and $\frac{1}{3}$ .

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

Step 1.3

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

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Step 1.3.1

Reorder terms.

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

Step 1.3.2

Remove parentheses.

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

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

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

$y = - 7 + 3 x$

Step 3.3

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.

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

Step 6

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

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Step 6.1

Combine $x$ and $\frac{1}{3}$ .

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

$- 3 x + y = - 7$

Step 6.2

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

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Step 6.2.1

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

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify $- 3 x - \frac{x}{3} + 3$ .

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Step 6.2.2.1.1

Remove parentheses.

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

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

Step 6.2.2.1.2

To write $- 3 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 3 x \cdot \frac{3}{3} - \frac{x}{3} + 3 = - 7$

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

Step 6.2.2.1.3

Simplify terms.

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Step 6.2.2.1.3.1

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

$\frac{- 3 x \cdot 3}{3} - \frac{x}{3} + 3 = - 7$

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

Step 6.2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{- 3 x \cdot 3 - x}{3} + 3 = - 7$

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

$\frac{- 3 x \cdot 3 - x}{3} + 3 = - 7$

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

Step 6.2.2.1.4

Simplify each term.

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Step 6.2.2.1.4.1

Simplify the numerator.

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Step 6.2.2.1.4.1.1

Factor $x$ out of $- 3 x \cdot 3 - x$ .

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Step 6.2.2.1.4.1.1.1

Factor $x$ out of $- 3 x \cdot 3$ .

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

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

Step 6.2.2.1.4.1.1.2

Factor $x$ out of $- x$ .

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

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

Step 6.2.2.1.4.1.1.3

Factor $x$ out of $x (- 3 \cdot 3) + x \cdot - 1$ .

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

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

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

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

Step 6.2.2.1.4.1.2

Multiply $- 3$ by $3$ .

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

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

Step 6.2.2.1.4.1.3

Subtract $1$ from $- 9$ .

$\frac{x \cdot - 10}{3} + 3 = - 7$

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

$\frac{x \cdot - 10}{3} + 3 = - 7$

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

Step 6.2.2.1.4.2

Move $- 10$ to the left of $x$ .

$\frac{- 10 \cdot x}{3} + 3 = - 7$

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

Step 6.2.2.1.4.3

Move the negative in front of the fraction.

$- \frac{10 x}{3} + 3 = - 7$

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

$- \frac{10 x}{3} + 3 = - 7$

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

$- \frac{10 x}{3} + 3 = - 7$

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

$- \frac{10 x}{3} + 3 = - 7$

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

$- \frac{10 x}{3} + 3 = - 7$

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

Step 6.3

Solve for $x$ in $- \frac{10 x}{3} + 3 = - 7$ .

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Step 6.3.1

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

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Step 6.3.1.1

Subtract $3$ from both sides of the equation.

$- \frac{10 x}{3} = - 7 - 3$

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

Step 6.3.1.2

Subtract $3$ from $- 7$ .

$- \frac{10 x}{3} = - 10$

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

$- \frac{10 x}{3} = - 10$

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

Step 6.3.2

Multiply both sides of the equation by $- \frac{3}{10}$ .

$- \frac{3}{10} \cdot (- \frac{10 x}{3}) = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3

Simplify both sides of the equation.

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Step 6.3.3.1

Simplify the left side.

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Step 6.3.3.1.1

Simplify $- \frac{3}{10} (- \frac{10 x}{3})$ .

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Step 6.3.3.1.1.1

Cancel the common factor of $3$ .

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Step 6.3.3.1.1.1.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$\frac{- 3}{10} \cdot (- \frac{10 x}{3}) = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.1.2

Move the leading negative in $- \frac{10 x}{3}$ into the numerator.

$\frac{- 3}{10} \cdot \frac{- 10 x}{3} = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{10} \cdot \frac{- 10 x}{3} = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{10} \cdot \frac{- 10 x}{3} = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{10} \cdot (- 10 x) = - \frac{3}{10} \cdot - 10$

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

$\frac{- 1}{10} \cdot (- 10 x) = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.2

Cancel the common factor of $10$ .

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Step 6.3.3.1.1.2.1

Factor $10$ out of $- 10 x$ .

$\frac{- 1}{10} \cdot (10 (- x)) = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{10} \cdot (10 (- x)) = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.2.3

Rewrite the expression.

$x = - \frac{3}{10} \cdot - 10$

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

$x = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.3

Multiply.

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Step 6.3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{10} \cdot - 10$

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

$x = - \frac{3}{10} \cdot - 10$

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

$x = - \frac{3}{10} \cdot - 10$

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

$x = - \frac{3}{10} \cdot - 10$

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

Step 6.3.3.2

Simplify the right side.

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Step 6.3.3.2.1

Simplify $- \frac{3}{10} \cdot - 10$ .

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Step 6.3.3.2.1.1

Cancel the common factor of $10$ .

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Step 6.3.3.2.1.1.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$x = \frac{- 3}{10} \cdot - 10$

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

Step 6.3.3.2.1.1.2

Factor $10$ out of $- 10$ .

$x = \frac{- 3}{10} \cdot (10 (- 1))$

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

Step 6.3.3.2.1.1.3

Cancel the common factor.

$x = \frac{- 3}{10} \cdot (10 \cdot - 1)$

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

Step 6.3.3.2.1.1.4

Rewrite the expression.

$x = - 3 \cdot - 1$

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

$x = - 3 \cdot - 1$

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

Step 6.3.3.2.1.2

Multiply $- 3$ by $- 1$ .

$x = 3$

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

$x = 3$

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

$x = 3$

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

$x = 3$

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

$x = 3$

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

Step 6.4

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

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Step 6.4.1

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

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

$x = 3$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $- \frac{3}{3} + 3$ .

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Step 6.4.2.1.1

Simplify each term.

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Step 6.4.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.4.2.1.1.1.1

Cancel the common factor.

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

$x = 3$

Step 6.4.2.1.1.1.2

Rewrite the expression.

$y = - 1 \cdot 1 + 3$

$x = 3$

$y = - 1 \cdot 1 + 3$

$x = 3$

Step 6.4.2.1.1.2

Multiply $- 1$ by $1$ .

$y = - 1 + 3$

$x = 3$

$y = - 1 + 3$

$x = 3$

Step 6.4.2.1.2

Add $- 1$ and $3$ .

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

Step 6.5

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

$(3, 2)$

Step 7

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

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

$m_{2} = 3$

$(3, 2)$

Step 8