Finite Math Examples

$y = x - 4$ ,

$y = - x + 7$

Step 1

Use the slope-intercept form to find the slope.

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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

Using the slope-intercept form, the slope is

$1$ .

$m_{1} = 1$

Step 2

Use the slope-intercept form to find the slope.

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Step 2.1

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 2.2

Using the slope-intercept form, the slope is

$- 1$ .

$m_{2} = - 1$

Step 3

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

$y = x - 4, y = - x + 7$

Step 4

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

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Step 4.1

Eliminate the equal sides of each equation and combine.

$x - 4 = - x + 7$

Step 4.2

Solve

$x - 4 = - x + 7$ for

$x$ .

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Step 4.2.1

Move all terms containing

$x$ to the left side of the equation.

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Step 4.2.1.1

Add

$x$ to both sides of the equation.

$x - 4 + x = 7$

Step 4.2.1.2

Add

$x$ and

$x$ .

$2 x - 4 = 7$

Step 4.2.2

Move all terms not containing

$x$ to the right side of the equation.

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Step 4.2.2.1

Add

$4$ to both sides of the equation.

$2 x = 7 + 4$

Step 4.2.2.2

Add

$7$ and

$4$ .

$2 x = 11$

Step 4.2.3

Divide each term in

$2 x = 11$ by

$2$ and simplify.

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Step 4.2.3.1

Divide each term in

$2 x = 11$ by

$2$ .

$\frac{2 x}{2} = \frac{11}{2}$

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of

$2$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{11}{2}$

Step 4.2.3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{11}{2}$

Step 4.3

Evaluate

$y$ when

$x = \frac{11}{2}$ .

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Step 4.3.1

Substitute

$\frac{11}{2}$ for

$x$ .

$y = - (\frac{11}{2}) + 7$

Step 4.3.2

Substitute

$\frac{11}{2}$ for

$x$ in

$y = - (\frac{11}{2}) + 7$ and solve for

$y$ .

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Step 4.3.2.1

Multiply

$- 1$ by

$\frac{11}{2}$ .

$y = - \frac{11}{2} + 7$

Step 4.3.2.2

Simplify

$- \frac{11}{2} + 7$ .

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Step 4.3.2.2.1

To write

$7$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y = - \frac{11}{2} + 7 \cdot \frac{2}{2}$

Step 4.3.2.2.2

Combine

$7$ and

$\frac{2}{2}$ .

$y = - \frac{11}{2} + \frac{7 \cdot 2}{2}$

Step 4.3.2.2.3

Combine the numerators over the common denominator.

$y = \frac{- 11 + 7 \cdot 2}{2}$

Step 4.3.2.2.4

Simplify the numerator.

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Step 4.3.2.2.4.1

Multiply

$7$ by

$2$ .

$y = \frac{- 11 + 14}{2}$

Step 4.3.2.2.4.2

Add

$- 11$ and

$14$ .

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

Step 4.4

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

$(\frac{11}{2}, \frac{3}{2})$

Step 5

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

$m_{1} = 1$

$m_{2} = - 1$

$(\frac{11}{2}, \frac{3}{2})$

Step 6