Finite Math Examples

$20 x + 10 y = 780$ , $10 x + 20 y = 720$

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 $20 x$ from both sides of the equation.

$10 y = 780 - 20 x$

Step 1.3

Divide each term in $10 y = 780 - 20 x$ by $10$ and simplify.

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Step 1.3.1

Divide each term in $10 y = 780 - 20 x$ by $10$ .

$\frac{10 y}{10} = \frac{780}{10} + \frac{- 20 x}{10}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $10$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{10 y}{10} = \frac{780}{10} + \frac{- 20 x}{10}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{780}{10} + \frac{- 20 x}{10}$

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 $780$ by $10$ .

$y = 78 + \frac{- 20 x}{10}$

Step 1.3.3.1.2

Cancel the common factor of $- 20$ and $10$ .

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Step 1.3.3.1.2.1

Factor $10$ out of $- 20 x$ .

$y = 78 + \frac{10 (- 2 x)}{10}$

Step 1.3.3.1.2.2

Cancel the common factors.

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Step 1.3.3.1.2.2.1

Factor $10$ out of $10$ .

$y = 78 + \frac{10 (- 2 x)}{10 (1)}$

Step 1.3.3.1.2.2.2

Cancel the common factor.

$y = 78 + \frac{10 (- 2 x)}{10 \cdot 1}$

Step 1.3.3.1.2.2.3

Rewrite the expression.

$y = 78 + \frac{- 2 x}{1}$

Step 1.3.3.1.2.2.4

Divide $- 2 x$ by $1$ .

$y = 78 - 2 x$

Step 1.4

Reorder $78$ and $- 2 x$ .

$y = - 2 x + 78$

Step 2

Using the slope-intercept form, the slope is $- 2$ .

$m_{1} = - 2$

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 $10 x$ from both sides of the equation.

$20 y = 720 - 10 x$

Step 3.3

Divide each term in $20 y = 720 - 10 x$ by $20$ and simplify.

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Step 3.3.1

Divide each term in $20 y = 720 - 10 x$ by $20$ .

$\frac{20 y}{20} = \frac{720}{20} + \frac{- 10 x}{20}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $20$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{20 y}{20} = \frac{720}{20} + \frac{- 10 x}{20}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{720}{20} + \frac{- 10 x}{20}$

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 $720$ by $20$ .

$y = 36 + \frac{- 10 x}{20}$

Step 3.3.3.1.2

Cancel the common factor of $- 10$ and $20$ .

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Step 3.3.3.1.2.1

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

$y = 36 + \frac{10 (- x)}{20}$

Step 3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.1.2.2.1

Factor $10$ out of $20$ .

$y = 36 + \frac{10 (- x)}{10 (2)}$

Step 3.3.3.1.2.2.2

Cancel the common factor.

$y = 36 + \frac{10 (- x)}{10 \cdot 2}$

Step 3.3.3.1.2.2.3

Rewrite the expression.

$y = 36 + \frac{- x}{2}$

Step 3.3.3.1.3

Move the negative in front of the fraction.

$y = 36 - \frac{x}{2}$

Step 3.4

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

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Step 3.4.1

Reorder $36$ and $- \frac{x}{2}$ .

$y = - \frac{x}{2} + 36$

Step 3.4.2

Reorder terms.

$y = - (\frac{1}{2} x) + 36$

Step 3.4.3

Remove parentheses.

$y = - \frac{1}{2} x + 36$

Step 4

Using the slope-intercept form, the slope is $- \frac{1}{2}$ .

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

Step 5

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

$20 x + 10 y = 780, 10 x + 20 y = 720$

Step 6

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

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Step 6.1

Solve for $x$ in $20 x + 10 y = 780$ .

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Step 6.1.1

Subtract $10 y$ from both sides of the equation.

$20 x = 780 - 10 y$

$10 x + 20 y = 720$

Step 6.1.2

Divide each term in $20 x = 780 - 10 y$ by $20$ and simplify.

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Step 6.1.2.1

Divide each term in $20 x = 780 - 10 y$ by $20$ .

$\frac{20 x}{20} = \frac{780}{20} + \frac{- 10 y}{20}$

$10 x + 20 y = 720$

Step 6.1.2.2

Simplify the left side.

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Step 6.1.2.2.1

Cancel the common factor of $20$ .

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Step 6.1.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} = \frac{780}{20} + \frac{- 10 y}{20}$

$10 x + 20 y = 720$

Step 6.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{780}{20} + \frac{- 10 y}{20}$

$10 x + 20 y = 720$

$x = \frac{780}{20} + \frac{- 10 y}{20}$

$10 x + 20 y = 720$

$x = \frac{780}{20} + \frac{- 10 y}{20}$

$10 x + 20 y = 720$

Step 6.1.2.3

Simplify the right side.

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Step 6.1.2.3.1

Simplify each term.

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Step 6.1.2.3.1.1

Divide $780$ by $20$ .

$x = 39 + \frac{- 10 y}{20}$

$10 x + 20 y = 720$

Step 6.1.2.3.1.2

Cancel the common factor of $- 10$ and $20$ .

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Step 6.1.2.3.1.2.1

Factor $10$ out of $- 10 y$ .

$x = 39 + \frac{10 (- y)}{20}$

$10 x + 20 y = 720$

Step 6.1.2.3.1.2.2

Cancel the common factors.

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Step 6.1.2.3.1.2.2.1

Factor $10$ out of $20$ .

$x = 39 + \frac{10 (- y)}{10 (2)}$

$10 x + 20 y = 720$

Step 6.1.2.3.1.2.2.2

Cancel the common factor.

$x = 39 + \frac{10 (- y)}{10 \cdot 2}$

$10 x + 20 y = 720$

Step 6.1.2.3.1.2.2.3

Rewrite the expression.

$x = 39 + \frac{- y}{2}$

$10 x + 20 y = 720$

$x = 39 + \frac{- y}{2}$

$10 x + 20 y = 720$

$x = 39 + \frac{- y}{2}$

$10 x + 20 y = 720$

Step 6.1.2.3.1.3

Move the negative in front of the fraction.

$x = 39 - \frac{y}{2}$

$10 x + 20 y = 720$

$x = 39 - \frac{y}{2}$

$10 x + 20 y = 720$

$x = 39 - \frac{y}{2}$

$10 x + 20 y = 720$

$x = 39 - \frac{y}{2}$

$10 x + 20 y = 720$

$x = 39 - \frac{y}{2}$

$10 x + 20 y = 720$

Step 6.2

Replace all occurrences of $x$ with $39 - \frac{y}{2}$ in each equation.

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Step 6.2.1

Replace all occurrences of $x$ in $10 x + 20 y = 720$ with $39 - \frac{y}{2}$ .

$10 (39 - \frac{y}{2}) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify $10 (39 - \frac{y}{2}) + 20 y$ .

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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.

$10 \cdot 39 + 10 (- \frac{y}{2}) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.1.2

Multiply $10$ by $39$ .

$390 + 10 (- \frac{y}{2}) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.1.3

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.3.1

Move the leading negative in $- \frac{y}{2}$ into the numerator.

$390 + 10 (\frac{- y}{2}) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.1.3.2

Factor $2$ out of $10$ .

$390 + 2 (5) (\frac{- y}{2}) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.1.3.3

Cancel the common factor.

$390 + 2 \cdot (5 (\frac{- y}{2})) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.1.3.4

Rewrite the expression.

$390 + 5 (- y) + 20 y = 720$

$x = 39 - \frac{y}{2}$

$390 + 5 (- y) + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.1.4

Multiply $- 1$ by $5$ .

$390 - 5 y + 20 y = 720$

$x = 39 - \frac{y}{2}$

$390 - 5 y + 20 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.2.2.1.2

Add $- 5 y$ and $20 y$ .

$390 + 15 y = 720$

$x = 39 - \frac{y}{2}$

$390 + 15 y = 720$

$x = 39 - \frac{y}{2}$

$390 + 15 y = 720$

$x = 39 - \frac{y}{2}$

$390 + 15 y = 720$

$x = 39 - \frac{y}{2}$

Step 6.3

Solve for $y$ in $390 + 15 y = 720$ .

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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 $390$ from both sides of the equation.

$15 y = 720 - 390$

$x = 39 - \frac{y}{2}$

Step 6.3.1.2

Subtract $390$ from $720$ .

$15 y = 330$

$x = 39 - \frac{y}{2}$

$15 y = 330$

$x = 39 - \frac{y}{2}$

Step 6.3.2

Divide each term in $15 y = 330$ by $15$ and simplify.

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Step 6.3.2.1

Divide each term in $15 y = 330$ by $15$ .

$\frac{15 y}{15} = \frac{330}{15}$

$x = 39 - \frac{y}{2}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $15$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{15 y}{15} = \frac{330}{15}$

$x = 39 - \frac{y}{2}$

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{330}{15}$

$x = 39 - \frac{y}{2}$

$y = \frac{330}{15}$

$x = 39 - \frac{y}{2}$

$y = \frac{330}{15}$

$x = 39 - \frac{y}{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $330$ by $15$ .

$y = 22$

$x = 39 - \frac{y}{2}$

$y = 22$

$x = 39 - \frac{y}{2}$

$y = 22$

$x = 39 - \frac{y}{2}$

$y = 22$

$x = 39 - \frac{y}{2}$

Step 6.4

Replace all occurrences of $y$ with $22$ in each equation.

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Step 6.4.1

Replace all occurrences of $y$ in $x = 39 - \frac{y}{2}$ with $22$ .

$x = 39 - \frac{22}{2}$

$y = 22$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $39 - \frac{22}{2}$ .

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Step 6.4.2.1.1

Simplify each term.

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Step 6.4.2.1.1.1

Divide $22$ by $2$ .

$x = 39 - 1 \cdot 11$

$y = 22$

Step 6.4.2.1.1.2

Multiply $- 1$ by $11$ .

$x = 39 - 11$

$y = 22$

$x = 39 - 11$

$y = 22$

Step 6.4.2.1.2

Subtract $11$ from $39$ .

$x = 28$

$y = 22$

$x = 28$

$y = 22$

$x = 28$

$y = 22$

$x = 28$

$y = 22$

Step 6.5

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

$(28, 22)$

Step 7

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

$m_{1} = - 2$

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

$(28, 22)$

Step 8