Finite Math Examples

$2 x + 5 y = 10$ , $x - 3 y = 3$

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 $2 x$ from both sides of the equation.

$5 y = 10 - 2 x$

Step 1.3

Divide each term in $5 y = 10 - 2 x$ by $5$ and simplify.

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Step 1.3.1

Divide each term in $5 y = 10 - 2 x$ by $5$ .

$\frac{5 y}{5} = \frac{10}{5} + \frac{- 2 x}{5}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $5$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{10}{5} + \frac{- 2 x}{5}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{10}{5} + \frac{- 2 x}{5}$

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 $10$ by $5$ .

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

Step 1.3.3.1.2

Move the negative in front of the fraction.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Reorder terms.

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

Step 1.4.3

Remove parentheses.

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

Step 2

Using the slope-intercept form, the slope is $- \frac{2}{5}$ .

$m_{1} = - \frac{2}{5}$

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.

$- 3 y = 3 - x$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

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 $3$ by $- 3$ .

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

Step 3.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 3.4

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

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Step 3.4.1

Reorder $- 1$ and $\frac{x}{3}$ .

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

Step 3.4.2

Reorder terms.

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

Step 4

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

$m_{2} = \frac{1}{3}$

Step 5

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

$2 x + 5 y = 10, x - 3 y = 3$

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.

$x = 3 + 3 y$

$2 x + 5 y = 10$

Step 6.2

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

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Step 6.2.1

Replace all occurrences of $x$ in $2 x + 5 y = 10$ with $3 + 3 y$ .

$2 (3 + 3 y) + 5 y = 10$

$x = 3 + 3 y$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify $2 (3 + 3 y) + 5 y$ .

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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.

$2 \cdot 3 + 2 (3 y) + 5 y = 10$

$x = 3 + 3 y$

Step 6.2.2.1.1.2

Multiply $2$ by $3$ .

$6 + 2 (3 y) + 5 y = 10$

$x = 3 + 3 y$

Step 6.2.2.1.1.3

Multiply $3$ by $2$ .

$6 + 6 y + 5 y = 10$

$x = 3 + 3 y$

$6 + 6 y + 5 y = 10$

$x = 3 + 3 y$

Step 6.2.2.1.2

Add $6 y$ and $5 y$ .

$6 + 11 y = 10$

$x = 3 + 3 y$

$6 + 11 y = 10$

$x = 3 + 3 y$

$6 + 11 y = 10$

$x = 3 + 3 y$

$6 + 11 y = 10$

$x = 3 + 3 y$

Step 6.3

Solve for $y$ in $6 + 11 y = 10$ .

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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 $6$ from both sides of the equation.

$11 y = 10 - 6$

$x = 3 + 3 y$

Step 6.3.1.2

Subtract $6$ from $10$ .

$11 y = 4$

$x = 3 + 3 y$

$11 y = 4$

$x = 3 + 3 y$

Step 6.3.2

Divide each term in $11 y = 4$ by $11$ and simplify.

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Step 6.3.2.1

Divide each term in $11 y = 4$ by $11$ .

$\frac{11 y}{11} = \frac{4}{11}$

$x = 3 + 3 y$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $11$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{11 y}{11} = \frac{4}{11}$

$x = 3 + 3 y$

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{11}$

$x = 3 + 3 y$

$y = \frac{4}{11}$

$x = 3 + 3 y$

$y = \frac{4}{11}$

$x = 3 + 3 y$

$y = \frac{4}{11}$

$x = 3 + 3 y$

$y = \frac{4}{11}$

$x = 3 + 3 y$

Step 6.4

Replace all occurrences of $y$ with $\frac{4}{11}$ in each equation.

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Step 6.4.1

Replace all occurrences of $y$ in $x = 3 + 3 y$ with $\frac{4}{11}$ .

$x = 3 + 3 (\frac{4}{11})$

$y = \frac{4}{11}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $3 + 3 (\frac{4}{11})$ .

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Step 6.4.2.1.1

Multiply $3 (\frac{4}{11})$ .

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Step 6.4.2.1.1.1

Combine $3$ and $\frac{4}{11}$ .

$x = 3 + \frac{3 \cdot 4}{11}$

$y = \frac{4}{11}$

Step 6.4.2.1.1.2

Multiply $3$ by $4$ .

$x = 3 + \frac{12}{11}$

$y = \frac{4}{11}$

$x = 3 + \frac{12}{11}$

$y = \frac{4}{11}$

Step 6.4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x = 3 \cdot \frac{11}{11} + \frac{12}{11}$

$y = \frac{4}{11}$

Step 6.4.2.1.3

Combine $3$ and $\frac{11}{11}$ .

$x = \frac{3 \cdot 11}{11} + \frac{12}{11}$

$y = \frac{4}{11}$

Step 6.4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{3 \cdot 11 + 12}{11}$

$y = \frac{4}{11}$

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Multiply $3$ by $11$ .

$x = \frac{33 + 12}{11}$

$y = \frac{4}{11}$

Step 6.4.2.1.5.2

Add $33$ and $12$ .

$x = \frac{45}{11}$

$y = \frac{4}{11}$

$x = \frac{45}{11}$

$y = \frac{4}{11}$

$x = \frac{45}{11}$

$y = \frac{4}{11}$

$x = \frac{45}{11}$

$y = \frac{4}{11}$

$x = \frac{45}{11}$

$y = \frac{4}{11}$

Step 6.5

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

$(\frac{45}{11}, \frac{4}{11})$

Step 7

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

$m_{1} = - \frac{2}{5}$

$m_{2} = \frac{1}{3}$

$(\frac{45}{11}, \frac{4}{11})$

Step 8