Finite Math Examples

$3 x + 4 y = 2$ , $4 x + 3 y = 4$

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 $3 x$ from both sides of the equation.

$4 y = 2 - 3 x$

Step 1.3

Divide each term in $4 y = 2 - 3 x$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 y = 2 - 3 x$ by $4$ .

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

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

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

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Step 1.3.3.1.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{4} + \frac{- 3 x}{4}$

Step 1.3.3.1.1.2

Cancel the common factors.

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Step 1.3.3.1.1.2.1

Factor $2$ out of $4$ .

$y = \frac{2 \cdot 1}{2 \cdot 2} + \frac{- 3 x}{4}$

Step 1.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot 2} + \frac{- 3 x}{4}$

Step 1.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{2} + \frac{- 3 x}{4}$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$y = \frac{1}{2} - \frac{3 x}{4}$

Step 1.4

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

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Step 1.4.1

Reorder $\frac{1}{2}$ and $- \frac{3 x}{4}$ .

$y = - \frac{3 x}{4} + \frac{1}{2}$

Step 1.4.2

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{1}{2}$

Step 1.4.3

Remove parentheses.

$y = - \frac{3}{4} x + \frac{1}{2}$

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

Subtract $4 x$ from both sides of the equation.

$3 y = 4 - 4 x$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

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

Step 3.4

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

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Step 3.4.1

Reorder $\frac{4}{3}$ and $- \frac{4 x}{3}$ .

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

Step 3.4.2

Reorder terms.

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

Step 3.4.3

Remove parentheses.

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

Step 4

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

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

Step 5

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

$3 x + 4 y = 2, 4 x + 3 y = 4$

Step 6

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

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Step 6.1

Solve for $x$ in $3 x + 4 y = 2$ .

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Step 6.1.1

Subtract $4 y$ from both sides of the equation.

$3 x = 2 - 4 y$

$4 x + 3 y = 4$

Step 6.1.2

Divide each term in $3 x = 2 - 4 y$ by $3$ and simplify.

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Step 6.1.2.1

Divide each term in $3 x = 2 - 4 y$ by $3$ .

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

$4 x + 3 y = 4$

Step 6.1.2.2

Simplify the left side.

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Step 6.1.2.2.1

Cancel the common factor of $3$ .

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Step 6.1.2.2.1.1

Cancel the common factor.

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

$4 x + 3 y = 4$

Step 6.1.2.2.1.2

Divide $x$ by $1$ .

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

$4 x + 3 y = 4$

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

$4 x + 3 y = 4$

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

$4 x + 3 y = 4$

Step 6.1.2.3

Simplify the right side.

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Step 6.1.2.3.1

Move the negative in front of the fraction.

$x = \frac{2}{3} - \frac{4 y}{3}$

$4 x + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$4 x + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$4 x + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$4 x + 3 y = 4$

Step 6.2

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

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Step 6.2.1

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

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

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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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.

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

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.1.2

Multiply $4 (\frac{2}{3})$ .

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Step 6.2.2.1.1.2.1

Combine $4$ and $\frac{2}{3}$ .

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

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.1.2.2

Multiply $4$ by $2$ .

$\frac{8}{3} + 4 (- \frac{4 y}{3}) + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$\frac{8}{3} + 4 (- \frac{4 y}{3}) + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.1.3

Multiply $4 (- \frac{4 y}{3})$ .

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Step 6.2.2.1.1.3.1

Multiply $- 1$ by $4$ .

$\frac{8}{3} - 4 \frac{4 y}{3} + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.1.3.2

Combine $- 4$ and $\frac{4 y}{3}$ .

$\frac{8}{3} + \frac{- 4 (4 y)}{3} + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.1.3.3

Multiply $4$ by $- 4$ .

$\frac{8}{3} + \frac{- 16 y}{3} + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$\frac{8}{3} + \frac{- 16 y}{3} + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.1.4

Move the negative in front of the fraction.

$\frac{8}{3} - \frac{16 y}{3} + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$\frac{8}{3} - \frac{16 y}{3} + 3 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.2

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - \frac{16 y}{3} + 3 y \cdot \frac{3}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.3

Combine $3 y$ and $\frac{3}{3}$ .

$\frac{8}{3} - \frac{16 y}{3} + \frac{3 y \cdot 3}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.4

Combine the numerators over the common denominator.

$\frac{8}{3} + \frac{- 16 y + 3 y \cdot 3}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.5

Combine the numerators over the common denominator.

$\frac{8 - 16 y + 3 y \cdot 3}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.6

Multiply $3$ by $3$ .

$\frac{8 - 16 y + 9 y}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.2.2.1.7

Add $- 16 y$ and $9 y$ .

$\frac{8 - 7 y}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$\frac{8 - 7 y}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$\frac{8 - 7 y}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$\frac{8 - 7 y}{3} = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3

Solve for $y$ in $\frac{8 - 7 y}{3} = 4$ .

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Step 6.3.1

Multiply both sides by $3$ .

$\frac{8 - 7 y}{3} \cdot 3 = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.2

Simplify.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Simplify $\frac{8 - 7 y}{3} \cdot 3$ .

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Step 6.3.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.3.2.1.1.1.1

Cancel the common factor.

$\frac{8 - 7 y}{3} \cdot 3 = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.2.1.1.1.2

Rewrite the expression.

$8 - 7 y = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

$8 - 7 y = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.2.1.1.2

Reorder $8$ and $- 7 y$ .

$- 7 y + 8 = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

$- 7 y + 8 = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

$- 7 y + 8 = 4 \cdot 3$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Multiply $4$ by $3$ .

$- 7 y + 8 = 12$

$x = \frac{2}{3} - \frac{4 y}{3}$

$- 7 y + 8 = 12$

$x = \frac{2}{3} - \frac{4 y}{3}$

$- 7 y + 8 = 12$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.3

Solve for $y$ .

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Step 6.3.3.1

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

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Step 6.3.3.1.1

Subtract $8$ from both sides of the equation.

$- 7 y = 12 - 8$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.3.1.2

Subtract $8$ from $12$ .

$- 7 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

$- 7 y = 4$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.3.2

Divide each term in $- 7 y = 4$ by $- 7$ and simplify.

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Step 6.3.3.2.1

Divide each term in $- 7 y = 4$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{4}{- 7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Cancel the common factor of $- 7$ .

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Step 6.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{4}{- 7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{- 7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

$y = \frac{4}{- 7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

$y = \frac{4}{- 7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{4}{7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

$y = - \frac{4}{7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

$y = - \frac{4}{7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

$y = - \frac{4}{7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

$y = - \frac{4}{7}$

$x = \frac{2}{3} - \frac{4 y}{3}$

Step 6.4

Replace all occurrences of $y$ with $- \frac{4}{7}$ in each equation.

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Step 6.4.1

Replace all occurrences of $y$ in $x = \frac{2}{3} - \frac{4 y}{3}$ with $- \frac{4}{7}$ .

$x = \frac{2}{3} - \frac{4 (- \frac{4}{7})}{3}$

$y = - \frac{4}{7}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $\frac{2}{3} - \frac{4 (- \frac{4}{7})}{3}$ .

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Step 6.4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{2 - 4 (- \frac{4}{7})}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.2

Multiply $- 4 (- \frac{4}{7})$ .

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Step 6.4.2.1.2.1

Multiply $- 1$ by $- 4$ .

$x = \frac{2 + 4 (\frac{4}{7})}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.2.2

Combine $4$ and $\frac{4}{7}$ .

$x = \frac{2 + \frac{4 \cdot 4}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.2.3

Multiply $4$ by $4$ .

$x = \frac{2 + \frac{16}{7}}{3}$

$y = - \frac{4}{7}$

$x = \frac{2 + \frac{16}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \frac{2 \cdot \frac{7}{7} + \frac{16}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.4

Combine $2$ and $\frac{7}{7}$ .

$x = \frac{\frac{2 \cdot 7}{7} + \frac{16}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{2 \cdot 7 + 16}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.6

Simplify the numerator.

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Step 6.4.2.1.6.1

Multiply $2$ by $7$ .

$x = \frac{\frac{14 + 16}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.6.2

Add $14$ and $16$ .

$x = \frac{\frac{30}{7}}{3}$

$y = - \frac{4}{7}$

$x = \frac{\frac{30}{7}}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{30}{7} \cdot \frac{1}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.8

Cancel the common factor of $3$ .

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Step 6.4.2.1.8.1

Factor $3$ out of $30$ .

$x = \frac{3 (10)}{7} \cdot \frac{1}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.8.2

Cancel the common factor.

$x = \frac{3 \cdot 10}{7} \cdot \frac{1}{3}$

$y = - \frac{4}{7}$

Step 6.4.2.1.8.3

Rewrite the expression.

$x = \frac{10}{7}$

$y = - \frac{4}{7}$

$x = \frac{10}{7}$

$y = - \frac{4}{7}$

$x = \frac{10}{7}$

$y = - \frac{4}{7}$

$x = \frac{10}{7}$

$y = - \frac{4}{7}$

$x = \frac{10}{7}$

$y = - \frac{4}{7}$

Step 6.5

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

$(\frac{10}{7}, - \frac{4}{7})$

Step 7

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

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

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

$(\frac{10}{7}, - \frac{4}{7})$

Step 8