Finite Math Examples

5 x + 2 y = 20

$5 x + 2 y = 20$ ,

x + 2 y = 8

$x + 2 y = 8$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 1.2

Subtract

5 x

$5 x$ from both sides of the equation.

2 y = 20 - 5 x

$2 y = 20 - 5 x$

Step 1.3

Divide each term in

2 y = 20 - 5 x

$2 y = 20 - 5 x$ by

2

$2$ and simplify.

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Step 1.3.1

Divide each term in

2 y = 20 - 5 x

$2 y = 20 - 5 x$ by

2

$2$ .

\frac{2 y}{2} = \frac{20}{2} + \frac{- 5 x}{2}

$\frac{2 y}{2} = \frac{20}{2} + \frac{- 5 x}{2}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of

2

$2$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{20}{2} + \frac{- 5 x}{2}$

Step 1.3.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{20}{2} + \frac{- 5 x}{2}$

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

$20$ by

$2$ .

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

Step 1.3.3.1.2

Move the negative in front of the fraction.

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

Step 1.4

Write in

$y = m x + b$ form.

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Step 1.4.1

Reorder

$10$ and

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

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

Step 1.4.2

Reorder terms.

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

Step 1.4.3

Remove parentheses.

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

Step 2

Using the slope-intercept form, the slope is

$- \frac{5}{2}$ .

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

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

$x$ from both sides of the equation.

$2 y = 8 - x$

Step 3.3

Divide each term in

$2 y = 8 - x$ by

$2$ and simplify.

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Step 3.3.1

Divide each term in

$2 y = 8 - x$ by

$2$ .

$\frac{2 y}{2} = \frac{8}{2} + \frac{- x}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

$2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{8}{2} + \frac{- x}{2}$

Step 3.3.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{8}{2} + \frac{- x}{2}$

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

$8$ by

$2$ .

$y = 4 + \frac{- x}{2}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

$y = 4 - \frac{x}{2}$

Step 3.4

Write in

$y = m x + b$ form.

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Step 3.4.1

Reorder

$4$ and

$- \frac{x}{2}$ .

$y = - \frac{x}{2} + 4$

Step 3.4.2

Reorder terms.

$y = - (\frac{1}{2} x) + 4$

Step 3.4.3

Remove parentheses.

$y = - \frac{1}{2} x + 4$

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.

$5 x + 2 y = 20, x + 2 y = 8$

Step 6

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

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Step 6.1

Subtract

$2 y$ from both sides of the equation.

$x = 8 - 2 y$

$5 x + 2 y = 20$

Step 6.2

Replace all occurrences of

$x$ with

$8 - 2 y$ in each equation.

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Step 6.2.1

Replace all occurrences of

$x$ in

$5 x + 2 y = 20$ with

$8 - 2 y$ .

$5 (8 - 2 y) + 2 y = 20$

$x = 8 - 2 y$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify

$5 (8 - 2 y) + 2 y$ .

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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.

$5 \cdot 8 + 5 (- 2 y) + 2 y = 20$

$x = 8 - 2 y$

Step 6.2.2.1.1.2

Multiply

$5$ by

$8$ .

$40 + 5 (- 2 y) + 2 y = 20$

$x = 8 - 2 y$

Step 6.2.2.1.1.3

Multiply

$- 2$ by

$5$ .

$40 - 10 y + 2 y = 20$

$x = 8 - 2 y$

$40 - 10 y + 2 y = 20$

$x = 8 - 2 y$

Step 6.2.2.1.2

Add

$- 10 y$ and

$2 y$ .

$40 - 8 y = 20$

$x = 8 - 2 y$

$40 - 8 y = 20$

$x = 8 - 2 y$

$40 - 8 y = 20$

$x = 8 - 2 y$

$40 - 8 y = 20$

$x = 8 - 2 y$

Step 6.3

Solve for

$y$ in

$40 - 8 y = 20$ .

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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

$40$ from both sides of the equation.

$- 8 y = 20 - 40$

$x = 8 - 2 y$

Step 6.3.1.2

Subtract

$40$ from

$20$ .

$- 8 y = - 20$

$x = 8 - 2 y$

$- 8 y = - 20$

$x = 8 - 2 y$

Step 6.3.2

Divide each term in

$- 8 y = - 20$ by

$- 8$ and simplify.

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Step 6.3.2.1

Divide each term in

$- 8 y = - 20$ by

$- 8$ .

$\frac{- 8 y}{- 8} = \frac{- 20}{- 8}$

$x = 8 - 2 y$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of

$- 8$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- 20}{- 8}$

$x = 8 - 2 y$

Step 6.3.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{- 20}{- 8}$

$x = 8 - 2 y$

$y = \frac{- 20}{- 8}$

$x = 8 - 2 y$

$y = \frac{- 20}{- 8}$

$x = 8 - 2 y$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Cancel the common factor of

$- 20$ and

$- 8$ .

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Step 6.3.2.3.1.1

Factor

$- 4$ out of

$- 20$ .

$y = \frac{- 4 \cdot 5}{- 8}$

$x = 8 - 2 y$

Step 6.3.2.3.1.2

Cancel the common factors.

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Step 6.3.2.3.1.2.1

Factor

$- 4$ out of

$- 8$ .

$y = \frac{- 4 \cdot 5}{- 4 \cdot 2}$

$x = 8 - 2 y$

Step 6.3.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 4 \cdot 5}{- 4 \cdot 2}$

$x = 8 - 2 y$

Step 6.3.2.3.1.2.3

Rewrite the expression.

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

$x = 8 - 2 y$

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

$x = 8 - 2 y$

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

$x = 8 - 2 y$

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

$x = 8 - 2 y$

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

$x = 8 - 2 y$

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

$x = 8 - 2 y$

Step 6.4

Replace all occurrences of

$y$ with

$\frac{5}{2}$ in each equation.

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Step 6.4.1

Replace all occurrences of

$y$ in

$x = 8 - 2 y$ with

$\frac{5}{2}$ .

$x = 8 - 2 (\frac{5}{2})$

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

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify

$8 - 2 (\frac{5}{2})$ .

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Step 6.4.2.1.1

Simplify each term.

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Step 6.4.2.1.1.1

Cancel the common factor of

$2$ .

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Step 6.4.2.1.1.1.1

Factor

$2$ out of

$- 2$ .

$x = 8 + 2 (- 1) (\frac{5}{2})$

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

Step 6.4.2.1.1.1.2

Cancel the common factor.

$x = 8 + 2 \cdot (- 1 (\frac{5}{2}))$

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

Step 6.4.2.1.1.1.3

Rewrite the expression.

$x = 8 - 1 \cdot 5$

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

$x = 8 - 1 \cdot 5$

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

Step 6.4.2.1.1.2

Multiply

$- 1$ by

$5$ .

$x = 8 - 5$

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

$x = 8 - 5$

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

Step 6.4.2.1.2

Subtract

$5$ from

$8$ .

$x = 3$

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

$x = 3$

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

$x = 3$

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

$x = 3$

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

Step 6.5

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

$(3, \frac{5}{2})$

Step 7

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

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

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

$(3, \frac{5}{2})$

Step 8