Finite Math Examples

2 x + 2 y = 3

$2 x + 2 y = 3$ ,

- x + 2 y = 1

$- x + 2 y = 1$

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

2 x

$2 x$ from both sides of the equation.

2 y = 3 - 2 x

$2 y = 3 - 2 x$

Step 1.3

Divide each term in

2 y = 3 - 2 x

$2 y = 3 - 2 x$ by

2

$2$ and simplify.

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Step 1.3.1

Divide each term in

2 y = 3 - 2 x

$2 y = 3 - 2 x$ by

2

$2$ .

\frac{2 y}{2} = \frac{3}{2} + \frac{- 2 x}{2}

$\frac{2 y}{2} = \frac{3}{2} + \frac{- 2 x}{2}$

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.

$\frac{2 y}{2} = \frac{3}{2} + \frac{- 2 x}{2}$

Step 1.3.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{3}{2} + \frac{- 2 x}{2}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Cancel the common factor of

$- 2$ and

$2$ .

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Step 1.3.3.1.1

Factor

$2$ out of

$- 2 x$ .

$y = \frac{3}{2} + \frac{2 (- x)}{2}$

Step 1.3.3.1.2

Cancel the common factors.

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Step 1.3.3.1.2.1

Factor

$2$ out of

$2$ .

$y = \frac{3}{2} + \frac{2 (- x)}{2 (1)}$

Step 1.3.3.1.2.2

Cancel the common factor.

$y = \frac{3}{2} + \frac{2 (- x)}{2 \cdot 1}$

Step 1.3.3.1.2.3

Rewrite the expression.

$y = \frac{3}{2} + \frac{- x}{1}$

Step 1.3.3.1.2.4

Divide

$- x$ by

$1$ .

$y = \frac{3}{2} - x$

Step 1.4

Reorder

$\frac{3}{2}$ and

$- x$ .

$y = - x + \frac{3}{2}$

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

Add

$x$ to both sides of the equation.

$2 y = 1 + x$

Step 3.3

Divide each term in

$2 y = 1 + x$ by

$2$ and simplify.

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Step 3.3.1

Divide each term in

$2 y = 1 + x$ by

$2$ .

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

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.

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

Step 3.3.2.1.2

Divide

$y$ by

$1$ .

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

Step 3.4

Write in

$y = m x + b$ form.

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Step 3.4.1

Reorder

$\frac{1}{2}$ and

$\frac{x}{2}$ .

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

Step 3.4.2

Reorder terms.

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

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.

$2 x + 2 y = 3, - x + 2 y = 1$

Step 6

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

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Step 6.1

Solve for

$x$ in

$2 x + 2 y = 3$ .

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Step 6.1.1

Subtract

$2 y$ from both sides of the equation.

$2 x = 3 - 2 y$

$- x + 2 y = 1$

Step 6.1.2

Divide each term in

$2 x = 3 - 2 y$ by

$2$ and simplify.

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Step 6.1.2.1

Divide each term in

$2 x = 3 - 2 y$ by

$2$ .

$\frac{2 x}{2} = \frac{3}{2} + \frac{- 2 y}{2}$

$- x + 2 y = 1$

Step 6.1.2.2

Simplify the left side.

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Step 6.1.2.2.1

Cancel the common factor of

$2$ .

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Step 6.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2} + \frac{- 2 y}{2}$

$- x + 2 y = 1$

Step 6.1.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{3}{2} + \frac{- 2 y}{2}$

$- x + 2 y = 1$

$x = \frac{3}{2} + \frac{- 2 y}{2}$

$- x + 2 y = 1$

$x = \frac{3}{2} + \frac{- 2 y}{2}$

$- x + 2 y = 1$

Step 6.1.2.3

Simplify the right side.

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Step 6.1.2.3.1

Cancel the common factor of

$- 2$ and

$2$ .

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Step 6.1.2.3.1.1

Factor

$2$ out of

$- 2 y$ .

$x = \frac{3}{2} + \frac{2 (- y)}{2}$

$- x + 2 y = 1$

Step 6.1.2.3.1.2

Cancel the common factors.

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Step 6.1.2.3.1.2.1

Factor

$2$ out of

$2$ .

$x = \frac{3}{2} + \frac{2 (- y)}{2 (1)}$

$- x + 2 y = 1$

Step 6.1.2.3.1.2.2

Cancel the common factor.

$x = \frac{3}{2} + \frac{2 (- y)}{2 \cdot 1}$

$- x + 2 y = 1$

Step 6.1.2.3.1.2.3

Rewrite the expression.

$x = \frac{3}{2} + \frac{- y}{1}$

$- x + 2 y = 1$

Step 6.1.2.3.1.2.4

Divide

$- y$ by

$1$ .

$x = \frac{3}{2} - y$

$- x + 2 y = 1$

$x = \frac{3}{2} - y$

$- x + 2 y = 1$

$x = \frac{3}{2} - y$

$- x + 2 y = 1$

$x = \frac{3}{2} - y$

$- x + 2 y = 1$

$x = \frac{3}{2} - y$

$- x + 2 y = 1$

$x = \frac{3}{2} - y$

$- x + 2 y = 1$

Step 6.2

Replace all occurrences of

$x$ with

$\frac{3}{2} - y$ in each equation.

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Step 6.2.1

Replace all occurrences of

$x$ in

$- x + 2 y = 1$ with

$\frac{3}{2} - y$ .

$- (\frac{3}{2} - y) + 2 y = 1$

$x = \frac{3}{2} - y$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify

$- (\frac{3}{2} - y) + 2 y$ .

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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.

$- \frac{3}{2} + y + 2 y = 1$

$x = \frac{3}{2} - y$

Step 6.2.2.1.1.2

Multiply

$- - y$ .

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Step 6.2.2.1.1.2.1

Multiply

$- 1$ by

$- 1$ .

$- \frac{3}{2} + 1 y + 2 y = 1$

$x = \frac{3}{2} - y$

Step 6.2.2.1.1.2.2

Multiply

$y$ by

$1$ .

$- \frac{3}{2} + y + 2 y = 1$

$x = \frac{3}{2} - y$

$- \frac{3}{2} + y + 2 y = 1$

$x = \frac{3}{2} - y$

$- \frac{3}{2} + y + 2 y = 1$

$x = \frac{3}{2} - y$

Step 6.2.2.1.2

Add

$y$ and

$2 y$ .

$- \frac{3}{2} + 3 y = 1$

$x = \frac{3}{2} - y$

$- \frac{3}{2} + 3 y = 1$

$x = \frac{3}{2} - y$

$- \frac{3}{2} + 3 y = 1$

$x = \frac{3}{2} - y$

$- \frac{3}{2} + 3 y = 1$

$x = \frac{3}{2} - y$

Step 6.3

Solve for

$y$ in

$- \frac{3}{2} + 3 y = 1$ .

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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

Add

$\frac{3}{2}$ to both sides of the equation.

$3 y = 1 + \frac{3}{2}$

$x = \frac{3}{2} - y$

Step 6.3.1.2

Write

$1$ as a fraction with a common denominator.

$3 y = \frac{2}{2} + \frac{3}{2}$

$x = \frac{3}{2} - y$

Step 6.3.1.3

Combine the numerators over the common denominator.

$3 y = \frac{2 + 3}{2}$

$x = \frac{3}{2} - y$

Step 6.3.1.4

Add

$2$ and

$3$ .

$3 y = \frac{5}{2}$

$x = \frac{3}{2} - y$

$3 y = \frac{5}{2}$

$x = \frac{3}{2} - y$

Step 6.3.2

Divide each term in

$3 y = \frac{5}{2}$ by

$3$ and simplify.

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Step 6.3.2.1

Divide each term in

$3 y = \frac{5}{2}$ by

$3$ .

$\frac{3 y}{3} = \frac{\frac{5}{2}}{3}$

$x = \frac{3}{2} - y$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of

$3$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\frac{5}{2}}{3}$

$x = \frac{3}{2} - y$

Step 6.3.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{\frac{5}{2}}{3}$

$x = \frac{3}{2} - y$

$y = \frac{\frac{5}{2}}{3}$

$x = \frac{3}{2} - y$

$y = \frac{\frac{5}{2}}{3}$

$x = \frac{3}{2} - y$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{5}{2} \cdot \frac{1}{3}$

$x = \frac{3}{2} - y$

Step 6.3.2.3.2

Multiply

$\frac{5}{2} \cdot \frac{1}{3}$ .

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Step 6.3.2.3.2.1

Multiply

$\frac{5}{2}$ by

$\frac{1}{3}$ .

$y = \frac{5}{2 \cdot 3}$

$x = \frac{3}{2} - y$

Step 6.3.2.3.2.2

Multiply

$2$ by

$3$ .

$y = \frac{5}{6}$

$x = \frac{3}{2} - y$

$y = \frac{5}{6}$

$x = \frac{3}{2} - y$

$y = \frac{5}{6}$

$x = \frac{3}{2} - y$

$y = \frac{5}{6}$

$x = \frac{3}{2} - y$

$y = \frac{5}{6}$

$x = \frac{3}{2} - y$

Step 6.4

Replace all occurrences of

$y$ with

$\frac{5}{6}$ in each equation.

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Step 6.4.1

Replace all occurrences of

$y$ in

$x = \frac{3}{2} - y$ with

$\frac{5}{6}$ .

$x = \frac{3}{2} - (\frac{5}{6})$

$y = \frac{5}{6}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify

$\frac{3}{2} - (\frac{5}{6})$ .

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Step 6.4.2.1.1

To write

$\frac{3}{2}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$x = \frac{3}{2} \cdot \frac{3}{3} - \frac{5}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.2

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 6.4.2.1.2.1

Multiply

$\frac{3}{2}$ by

$\frac{3}{3}$ .

$x = \frac{3 \cdot 3}{2 \cdot 3} - \frac{5}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.2.2

Multiply

$2$ by

$3$ .

$x = \frac{3 \cdot 3}{6} - \frac{5}{6}$

$y = \frac{5}{6}$

$x = \frac{3 \cdot 3}{6} - \frac{5}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.3

Combine the numerators over the common denominator.

$x = \frac{3 \cdot 3 - 5}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.4

Simplify the numerator.

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Step 6.4.2.1.4.1

Multiply

$3$ by

$3$ .

$x = \frac{9 - 5}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.4.2

Subtract

$5$ from

$9$ .

$x = \frac{4}{6}$

$y = \frac{5}{6}$

$x = \frac{4}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.5

Cancel the common factor of

$4$ and

$6$ .

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Step 6.4.2.1.5.1

Factor

$2$ out of

$4$ .

$x = \frac{2 (2)}{6}$

$y = \frac{5}{6}$

Step 6.4.2.1.5.2

Cancel the common factors.

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Step 6.4.2.1.5.2.1

Factor

$2$ out of

$6$ .

$x = \frac{2 \cdot 2}{2 \cdot 3}$

$y = \frac{5}{6}$

Step 6.4.2.1.5.2.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{2 \cdot 3}$

$y = \frac{5}{6}$

Step 6.4.2.1.5.2.3

Rewrite the expression.

$x = \frac{2}{3}$

$y = \frac{5}{6}$

$x = \frac{2}{3}$

$y = \frac{5}{6}$

$x = \frac{2}{3}$

$y = \frac{5}{6}$

$x = \frac{2}{3}$

$y = \frac{5}{6}$

$x = \frac{2}{3}$

$y = \frac{5}{6}$

$x = \frac{2}{3}$

$y = \frac{5}{6}$

Step 6.5

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

$(\frac{2}{3}, \frac{5}{6})$

Step 7

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

$m_{1} = - 1$

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

$(\frac{2}{3}, \frac{5}{6})$

Step 8