Finite Math Examples

$4 y = 3 x + 16$ , $- 4 y = 3 x - 12$

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

Divide each term in $4 y = 3 x + 16$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 y = 3 x + 16$ by $4$ .

$\frac{4 y}{4} = \frac{3 x}{4} + \frac{16}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{3 x}{4} + \frac{16}{4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3 x}{4} + \frac{16}{4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $16$ by $4$ .

$y = \frac{3 x}{4} + 4$

Step 1.3

Reorder terms.

$y = \frac{3}{4} x + 4$

Step 2

Using the slope-intercept form, the slope is $\frac{3}{4}$ .

$m_{1} = \frac{3}{4}$

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

Divide each term in $- 4 y = 3 x - 12$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 y = 3 x - 12$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{3 x}{- 4} + \frac{- 12}{- 4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{3 x}{- 4} + \frac{- 12}{- 4}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3 x}{- 4} + \frac{- 12}{- 4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{3 x}{4} + \frac{- 12}{- 4}$

Step 3.2.3.1.2

Divide $- 12$ by $- 4$ .

$y = - \frac{3 x}{4} + 3$

Step 3.3

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

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Step 3.3.1

Reorder terms.

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

Step 3.3.2

Remove parentheses.

$y = - \frac{3}{4} x + 3$

Step 4

Using the slope-intercept form, the slope is $- \frac{3}{4}$ .

$m_{2} = - \frac{3}{4}$

Step 5

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

$4 y = 3 x + 16, - 4 y = 3 x - 12$

Step 6

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

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Step 6.1

Divide each term in $4 y = 3 x + 16$ by $4$ and simplify.

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Step 6.1.1

Divide each term in $4 y = 3 x + 16$ by $4$ .

$\frac{4 y}{4} = \frac{3 x}{4} + \frac{16}{4}$

$- 4 y = 3 x - 12$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $4$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{3 x}{4} + \frac{16}{4}$

$- 4 y = 3 x - 12$

Step 6.1.2.1.2

Divide $y$ by $1$ .

$y = \frac{3 x}{4} + \frac{16}{4}$

$- 4 y = 3 x - 12$

$y = \frac{3 x}{4} + \frac{16}{4}$

$- 4 y = 3 x - 12$

$y = \frac{3 x}{4} + \frac{16}{4}$

$- 4 y = 3 x - 12$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $16$ by $4$ .

$y = \frac{3 x}{4} + 4$

$- 4 y = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 4 y = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 4 y = 3 x - 12$

Step 6.2

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

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Step 6.2.1

Replace all occurrences of $y$ in $- 4 y = 3 x - 12$ with $\frac{3 x}{4} + 4$ .

$- 4 (\frac{3 x}{4} + 4) = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Apply the distributive property.

$- 4 \frac{3 x}{4} - 4 \cdot 4 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.2.2.1.2

Cancel the common factor of $4$ .

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Step 6.2.2.1.2.1

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{3 x}{4}) - 4 \cdot 4 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.2.2.1.2.2

Cancel the common factor.

$4 \cdot (- 1 \frac{3 x}{4}) - 4 \cdot 4 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.2.2.1.2.3

Rewrite the expression.

$- 1 (3 x) - 4 \cdot 4 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 1 (3 x) - 4 \cdot 4 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.2.2.1.3

Multiply.

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Step 6.2.2.1.3.1

Multiply $3$ by $- 1$ .

$- 3 x - 4 \cdot 4 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.2.2.1.3.2

Multiply $- 4$ by $4$ .

$- 3 x - 16 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 3 x - 16 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 3 x - 16 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 3 x - 16 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

$- 3 x - 16 = 3 x - 12$

$y = \frac{3 x}{4} + 4$

Step 6.3

Solve for $x$ in $- 3 x - 16 = 3 x - 12$ .

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Step 6.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 6.3.1.1

Subtract $3 x$ from both sides of the equation.

$- 3 x - 16 - 3 x = - 12$

$y = \frac{3 x}{4} + 4$

Step 6.3.1.2

Subtract $3 x$ from $- 3 x$ .

$- 6 x - 16 = - 12$

$y = \frac{3 x}{4} + 4$

$- 6 x - 16 = - 12$

$y = \frac{3 x}{4} + 4$

Step 6.3.2

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

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Step 6.3.2.1

Add $16$ to both sides of the equation.

$- 6 x = - 12 + 16$

$y = \frac{3 x}{4} + 4$

Step 6.3.2.2

Add $- 12$ and $16$ .

$- 6 x = 4$

$y = \frac{3 x}{4} + 4$

$- 6 x = 4$

$y = \frac{3 x}{4} + 4$

Step 6.3.3

Divide each term in $- 6 x = 4$ by $- 6$ and simplify.

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Step 6.3.3.1

Divide each term in $- 6 x = 4$ by $- 6$ .

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 6.3.3.2.1.1

Cancel the common factor.

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.2.1.2

Divide $x$ by $1$ .

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Cancel the common factor of $4$ and $- 6$ .

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Step 6.3.3.3.1.1

Factor $2$ out of $4$ .

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.3.1.2

Cancel the common factors.

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Step 6.3.3.3.1.2.1

Factor $2$ out of $- 6$ .

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.3.1.2.2

Cancel the common factor.

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.3.1.2.3

Rewrite the expression.

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

Step 6.3.3.3.2

Move the negative in front of the fraction.

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

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

$y = \frac{3 x}{4} + 4$

Step 6.4

Replace all occurrences of $x$ with $- \frac{2}{3}$ in each equation.

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Step 6.4.1

Replace all occurrences of $x$ in $y = \frac{3 x}{4} + 4$ with $- \frac{2}{3}$ .

$y = \frac{3 (- \frac{2}{3})}{4} + 4$

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

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $\frac{3 (- \frac{2}{3})}{4} + 4$ .

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Step 6.4.2.1.1

Simplify each term.

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Step 6.4.2.1.1.1

Simplify the numerator.

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Step 6.4.2.1.1.1.1

Multiply $- 1$ by $3$ .

$y = \frac{- 3 (\frac{2}{3})}{4} + 4$

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

Step 6.4.2.1.1.1.2

Combine $- 3$ and $\frac{2}{3}$ .

$y = \frac{\frac{- 3 \cdot 2}{3}}{4} + 4$

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

$y = \frac{\frac{- 3 \cdot 2}{3}}{4} + 4$

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

Step 6.4.2.1.1.2

Multiply $- 3$ by $2$ .

$y = \frac{\frac{- 6}{3}}{4} + 4$

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

Step 6.4.2.1.1.3

Divide $- 6$ by $3$ .

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

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

Step 6.4.2.1.1.4

Cancel the common factor of $- 2$ and $4$ .

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Step 6.4.2.1.1.4.1

Factor $2$ out of $- 2$ .

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

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

Step 6.4.2.1.1.4.2

Cancel the common factors.

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Step 6.4.2.1.1.4.2.1

Factor $2$ out of $4$ .

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

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

Step 6.4.2.1.1.4.2.2

Cancel the common factor.

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

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

Step 6.4.2.1.1.4.2.3

Rewrite the expression.

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

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

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

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

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

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

Step 6.4.2.1.1.5

Move the negative in front of the fraction.

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

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

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

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

Step 6.4.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

Step 6.4.2.1.3

Combine $4$ and $\frac{2}{2}$ .

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

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

Step 6.4.2.1.4

Combine the numerators over the common denominator.

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

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

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Multiply $4$ by $2$ .

$y = \frac{- 1 + 8}{2}$

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

Step 6.4.2.1.5.2

Add $- 1$ and $8$ .

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

Step 6.5

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

$(- \frac{2}{3}, \frac{7}{2})$

Step 7

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

$m_{1} = \frac{3}{4}$

$m_{2} = - \frac{3}{4}$

$(- \frac{2}{3}, \frac{7}{2})$

Step 8