Finite Math Examples

$x + y = 11$ , $x - y = 5$

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.

$y = 11 - x$

Step 1.3

Reorder $11$ and $- x$ .

$y = - x + 11$

Step 2

Using the slope-intercept form, the slope is $- 1$ .

$m_{1} = - 1$

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 $x$ from both sides of the equation.

$- y = 5 - x$

Step 3.3

Divide each term in $- y = 5 - x$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- y = 5 - x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{5}{- 1} + \frac{- x}{- 1}$

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.

$\frac{y}{1} = \frac{5}{- 1} + \frac{- x}{- 1}$

Step 3.3.2.2

Divide $y$ by $1$ .

$y = \frac{5}{- 1} + \frac{- x}{- 1}$

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 $5$ by $- 1$ .

$y = - 5 + \frac{- x}{- 1}$

Step 3.3.3.1.2

Dividing two negative values results in a positive value.

$y = - 5 + \frac{x}{1}$

Step 3.3.3.1.3

Divide $x$ by $1$ .

$y = - 5 + x$

Step 3.4

Reorder $- 5$ and $x$ .

$y = x - 5$

Step 4

Using the slope-intercept form, the slope is $1$ .

$m_{2} = 1$

Step 5

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

$x + y = 11, x - y = 5$

Step 6

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

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Step 6.1

Subtract $y$ from both sides of the equation.

$x = 11 - y$

$x - y = 5$

Step 6.2

Replace all occurrences of $x$ with $11 - y$ in each equation.

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Step 6.2.1

Replace all occurrences of $x$ in $x - y = 5$ with $11 - y$ .

$(11 - y) - y = 5$

$x = 11 - y$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Subtract $y$ from $- y$ .

$11 - 2 y = 5$

$x = 11 - y$

$11 - 2 y = 5$

$x = 11 - y$

$11 - 2 y = 5$

$x = 11 - y$

Step 6.3

Solve for $y$ in $11 - 2 y = 5$ .

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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 $11$ from both sides of the equation.

$- 2 y = 5 - 11$

$x = 11 - y$

Step 6.3.1.2

Subtract $11$ from $5$ .

$- 2 y = - 6$

$x = 11 - y$

$- 2 y = - 6$

$x = 11 - y$

Step 6.3.2

Divide each term in $- 2 y = - 6$ by $- 2$ and simplify.

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Step 6.3.2.1

Divide each term in $- 2 y = - 6$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 6}{- 2}$

$x = 11 - y$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 6}{- 2}$

$x = 11 - y$

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 6}{- 2}$

$x = 11 - y$

$y = \frac{- 6}{- 2}$

$x = 11 - y$

$y = \frac{- 6}{- 2}$

$x = 11 - y$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $- 6$ by $- 2$ .

$y = 3$

$x = 11 - y$

$y = 3$

$x = 11 - y$

$y = 3$

$x = 11 - y$

$y = 3$

$x = 11 - y$

Step 6.4

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

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Step 6.4.1

Replace all occurrences of $y$ in $x = 11 - y$ with $3$ .

$x = 11 - (3)$

$y = 3$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $11 - (3)$ .

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Step 6.4.2.1.1

Multiply $- 1$ by $3$ .

$x = 11 - 3$

$y = 3$

Step 6.4.2.1.2

Subtract $3$ from $11$ .

$x = 8$

$y = 3$

$x = 8$

$y = 3$

$x = 8$

$y = 3$

$x = 8$

$y = 3$

Step 6.5

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

$(8, 3)$

Step 7

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

$m_{1} = - 1$

$m_{2} = 1$

$(8, 3)$

Step 8