Finite Math Examples

$2 x + y = 20$ , $10 x + y = 36$ , $2 x + 5 y = 36$

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.

$y = 20 - 2 x$

Step 1.3

Reorder $20$ and $- 2 x$ .

$y = - 2 x + 20$

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.

$y = 36 - 10 x$

Step 3.3

Reorder $36$ and $- 10 x$ .

$y = - 10 x + 36$

Step 4

Using the slope-intercept form, the slope is $- 10$ .

$m_{2} = - 10$

Step 5

Rewrite in slope-intercept form.

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Step 5.1

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 5.2

Subtract $2 x$ from both sides of the equation.

$5 y = 36 - 2 x$

Step 5.3

Divide each term in $5 y = 36 - 2 x$ by $5$ and simplify.

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Step 5.3.1

Divide each term in $5 y = 36 - 2 x$ by $5$ .

$\frac{5 y}{5} = \frac{36}{5} + \frac{- 2 x}{5}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $5$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{36}{5} + \frac{- 2 x}{5}$

Step 5.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{36}{5} + \frac{- 2 x}{5}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

$y = \frac{36}{5} - \frac{2 x}{5}$

Step 5.4

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

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Step 5.4.1

Reorder $\frac{36}{5}$ and $- \frac{2 x}{5}$ .

$y = - \frac{2 x}{5} + \frac{36}{5}$

Step 5.4.2

Reorder terms.

$y = - (\frac{2}{5} x) + \frac{36}{5}$

Step 5.4.3

Remove parentheses.

$y = - \frac{2}{5} x + \frac{36}{5}$

Step 6

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

$m_{3} = - \frac{2}{5}$

Step 7

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

$2 x + y = 20, 10 x + y = 36, 2 x + 5 y = 36$

Step 8

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

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Step 8.1

Subtract $2 x$ from both sides of the equation.

$y = 20 - 2 x$

$10 x + y = 36$

$2 x + 5 y = 36$

Step 8.2

Replace all occurrences of $y$ with $20 - 2 x$ in each equation.

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Step 8.2.1

Replace all occurrences of $y$ in $10 x + y = 36$ with $20 - 2 x$ .

$10 x + 20 - 2 x = 36$

$y = 20 - 2 x$

$2 x + 5 y = 36$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Simplify $10 x + 20 - 2 x$ .

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Step 8.2.2.1.1

Remove parentheses.

$10 x + 20 - 2 x = 36$

$y = 20 - 2 x$

$2 x + 5 y = 36$

Step 8.2.2.1.2

Subtract $2 x$ from $10 x$ .

$8 x + 20 = 36$

$y = 20 - 2 x$

$2 x + 5 y = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

$2 x + 5 y = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

$2 x + 5 y = 36$

Step 8.2.3

Replace all occurrences of $y$ in $2 x + 5 y = 36$ with $20 - 2 x$ .

$2 x + 5 (20 - 2 x) = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.2.4

Simplify the left side.

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Step 8.2.4.1

Simplify $2 x + 5 (20 - 2 x)$ .

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Step 8.2.4.1.1

Simplify each term.

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Step 8.2.4.1.1.1

Apply the distributive property.

$2 x + 5 \cdot 20 + 5 (- 2 x) = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.2.4.1.1.2

Multiply $5$ by $20$ .

$2 x + 100 + 5 (- 2 x) = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.2.4.1.1.3

Multiply $- 2$ by $5$ .

$2 x + 100 - 10 x = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

$2 x + 100 - 10 x = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.2.4.1.2

Subtract $10 x$ from $2 x$ .

$- 8 x + 100 = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

$- 8 x + 100 = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

$- 8 x + 100 = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

$- 8 x + 100 = 36$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.3

Solve for $x$ in $- 8 x + 100 = 36$ .

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Step 8.3.1

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

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Step 8.3.1.1

Subtract $100$ from both sides of the equation.

$- 8 x = 36 - 100$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.3.1.2

Subtract $100$ from $36$ .

$- 8 x = - 64$

$8 x + 20 = 36$

$y = 20 - 2 x$

$- 8 x = - 64$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.3.2

Divide each term in $- 8 x = - 64$ by $- 8$ and simplify.

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Step 8.3.2.1

Divide each term in $- 8 x = - 64$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 64}{- 8}$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.3.2.2

Simplify the left side.

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Step 8.3.2.2.1

Cancel the common factor of $- 8$ .

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Step 8.3.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 64}{- 8}$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 64}{- 8}$

$8 x + 20 = 36$

$y = 20 - 2 x$

$x = \frac{- 64}{- 8}$

$8 x + 20 = 36$

$y = 20 - 2 x$

$x = \frac{- 64}{- 8}$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.3.2.3

Simplify the right side.

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Step 8.3.2.3.1

Divide $- 64$ by $- 8$ .

$x = 8$

$8 x + 20 = 36$

$y = 20 - 2 x$

$x = 8$

$8 x + 20 = 36$

$y = 20 - 2 x$

$x = 8$

$8 x + 20 = 36$

$y = 20 - 2 x$

$x = 8$

$8 x + 20 = 36$

$y = 20 - 2 x$

Step 8.4

Replace all occurrences of $x$ with $8$ in each equation.

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Step 8.4.1

Replace all occurrences of $x$ in $8 x + 20 = 36$ with $8$ .

$8 (8) + 20 = 36$

$x = 8$

$y = 20 - 2 x$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Simplify $8 (8) + 20$ .

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Step 8.4.2.1.1

Multiply $8$ by $8$ .

$64 + 20 = 36$

$x = 8$

$y = 20 - 2 x$

Step 8.4.2.1.2

Add $64$ and $20$ .

$84 = 36$

$x = 8$

$y = 20 - 2 x$

$84 = 36$

$x = 8$

$y = 20 - 2 x$

$84 = 36$

$x = 8$

$y = 20 - 2 x$

Step 8.4.3

Replace all occurrences of $x$ in $y = 20 - 2 x$ with $8$ .

$y = 20 - 2 \cdot 8$

$84 = 36$

$x = 8$

Step 8.4.4

Simplify the right side.

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Step 8.4.4.1

Simplify $20 - 2 \cdot 8$ .

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Step 8.4.4.1.1

Multiply $- 2$ by $8$ .

$y = 20 - 16$

$84 = 36$

$x = 8$

Step 8.4.4.1.2

Subtract $16$ from $20$ .

$y = 4$

$84 = 36$

$x = 8$

$y = 4$

$84 = 36$

$x = 8$

$y = 4$

$84 = 36$

$x = 8$

$y = 4$

$84 = 36$

$x = 8$

Step 8.5

Since $84 = 36$ is not true, there is no solution.

No solution

Step 9

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

No solution

Step 10