Finite Math Examples

$- 7 x + 3 y = 4$ , $4 x - 5 y = 24$

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

Add $7 x$ to both sides of the equation.

$3 y = 4 + 7 x$

Step 1.3

Divide each term in $3 y = 4 + 7 x$ by $3$ and simplify.

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Step 1.3.1

Divide each term in $3 y = 4 + 7 x$ by $3$ .

$\frac{3 y}{3} = \frac{4}{3} + \frac{7 x}{3}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{4}{3} + \frac{7 x}{3}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{3} + \frac{7 x}{3}$

Step 1.4

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

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Step 1.4.1

Reorder $\frac{4}{3}$ and $\frac{7 x}{3}$ .

$y = \frac{7 x}{3} + \frac{4}{3}$

Step 1.4.2

Reorder terms.

$y = \frac{7}{3} x + \frac{4}{3}$

Step 2

Using the slope-intercept form, the slope is $\frac{7}{3}$ .

$m_{1} = \frac{7}{3}$

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.

$- 5 y = 24 - 4 x$

Step 3.3

Divide each term in $- 5 y = 24 - 4 x$ by $- 5$ and simplify.

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Step 3.3.1

Divide each term in $- 5 y = 24 - 4 x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{24}{- 5} + \frac{- 4 x}{- 5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{24}{- 5} + \frac{- 4 x}{- 5}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{24}{- 5} + \frac{- 4 x}{- 5}$

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

Move the negative in front of the fraction.

$y = - \frac{24}{5} + \frac{- 4 x}{- 5}$

Step 3.3.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{24}{5} + \frac{4 x}{5}$

Step 3.4

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

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Step 3.4.1

Reorder $- \frac{24}{5}$ and $\frac{4 x}{5}$ .

$y = \frac{4 x}{5} - \frac{24}{5}$

Step 3.4.2

Reorder terms.

$y = \frac{4}{5} x - \frac{24}{5}$

Step 4

Using the slope-intercept form, the slope is $\frac{4}{5}$ .

$m_{2} = \frac{4}{5}$

Step 5

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

$- 7 x + 3 y = 4, 4 x - 5 y = 24$

Step 6

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

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Step 6.1

Solve for $x$ in $- 7 x + 3 y = 4$ .

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Step 6.1.1

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

$- 7 x = 4 - 3 y$

$4 x - 5 y = 24$

Step 6.1.2

Divide each term in $- 7 x = 4 - 3 y$ by $- 7$ and simplify.

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Step 6.1.2.1

Divide each term in $- 7 x = 4 - 3 y$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{4}{- 7} + \frac{- 3 y}{- 7}$

$4 x - 5 y = 24$

Step 6.1.2.2

Simplify the left side.

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Step 6.1.2.2.1

Cancel the common factor of $- 7$ .

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Step 6.1.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{4}{- 7} + \frac{- 3 y}{- 7}$

$4 x - 5 y = 24$

Step 6.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{- 7} + \frac{- 3 y}{- 7}$

$4 x - 5 y = 24$

$x = \frac{4}{- 7} + \frac{- 3 y}{- 7}$

$4 x - 5 y = 24$

$x = \frac{4}{- 7} + \frac{- 3 y}{- 7}$

$4 x - 5 y = 24$

Step 6.1.2.3

Simplify the right side.

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Step 6.1.2.3.1

Simplify each term.

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Step 6.1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{4}{7} + \frac{- 3 y}{- 7}$

$4 x - 5 y = 24$

Step 6.1.2.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{4}{7} + \frac{3 y}{7}$

$4 x - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$4 x - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$4 x - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$4 x - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$4 x - 5 y = 24$

Step 6.2

Replace all occurrences of $x$ with $- \frac{4}{7} + \frac{3 y}{7}$ in each equation.

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Step 6.2.1

Replace all occurrences of $x$ in $4 x - 5 y = 24$ with $- \frac{4}{7} + \frac{3 y}{7}$ .

$4 (- \frac{4}{7} + \frac{3 y}{7}) - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify $4 (- \frac{4}{7} + \frac{3 y}{7}) - 5 y$ .

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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.

$4 (- \frac{4}{7}) + 4 (\frac{3 y}{7}) - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.1.2

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

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Step 6.2.2.1.1.2.1

Multiply $- 1$ by $4$ .

$- 4 (\frac{4}{7}) + 4 (\frac{3 y}{7}) - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.1.2.2

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

$\frac{- 4 \cdot 4}{7} + 4 (\frac{3 y}{7}) - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.1.2.3

Multiply $- 4$ by $4$ .

$\frac{- 16}{7} + 4 (\frac{3 y}{7}) - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$\frac{- 16}{7} + 4 (\frac{3 y}{7}) - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.1.3

Multiply $4 \frac{3 y}{7}$ .

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Step 6.2.2.1.1.3.1

Combine $4$ and $\frac{3 y}{7}$ .

$\frac{- 16}{7} + \frac{4 (3 y)}{7} - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.1.3.2

Multiply $3$ by $4$ .

$\frac{- 16}{7} + \frac{12 y}{7} - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$\frac{- 16}{7} + \frac{12 y}{7} - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{16}{7} + \frac{12 y}{7} - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$- \frac{16}{7} + \frac{12 y}{7} - 5 y = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.2

To write $- 5 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{16}{7} + \frac{12 y}{7} - 5 y \cdot \frac{7}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.3

Combine $- 5 y$ and $\frac{7}{7}$ .

$- \frac{16}{7} + \frac{12 y}{7} + \frac{- 5 y \cdot 7}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.4

Combine the numerators over the common denominator.

$- \frac{16}{7} + \frac{12 y - 5 y \cdot 7}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 16 + 12 y - 5 y \cdot 7}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.6

Multiply $7$ by $- 5$ .

$\frac{- 16 + 12 y - 35 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.7

Subtract $35 y$ from $12 y$ .

$\frac{- 16 - 23 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.8

Rewrite $- 16$ as $- 1 (16)$ .

$\frac{- 1 \cdot 16 - 23 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.9

Factor $- 1$ out of $- 23 y$ .

$\frac{- 1 \cdot 16 - (23 y)}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.10

Factor $- 1$ out of $- 1 (16) - (23 y)$ .

$\frac{- 1 (16 + 23 y)}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.2.2.1.11

Move the negative in front of the fraction.

$- \frac{16 + 23 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$- \frac{16 + 23 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$- \frac{16 + 23 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$- \frac{16 + 23 y}{7} = 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3

Solve for $y$ in $- \frac{16 + 23 y}{7} = 24$ .

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Step 6.3.1

Multiply both sides of the equation by $- 7$ .

$- 7 (- \frac{16 + 23 y}{7}) = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2

Simplify both sides of the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Simplify $- 7 (- \frac{16 + 23 y}{7})$ .

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Step 6.3.2.1.1.1

Cancel the common factor of $7$ .

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Step 6.3.2.1.1.1.1

Move the leading negative in $- \frac{16 + 23 y}{7}$ into the numerator.

$- 7 \frac{- (16 + 23 y)}{7} = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2.1.1.1.2

Factor $7$ out of $- 7$ .

$7 (- 1) (\frac{- (16 + 23 y)}{7}) = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2.1.1.1.3

Cancel the common factor.

$7 \cdot (- 1 \frac{- (16 + 23 y)}{7}) = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2.1.1.1.4

Rewrite the expression.

$16 + 23 y = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$16 + 23 y = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2.1.1.2

Multiply.

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Step 6.3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (16 + 23 y) = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2.1.1.2.2

Multiply $16 + 23 y$ by $1$ .

$16 + 23 y = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$16 + 23 y = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$16 + 23 y = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$16 + 23 y = - 7 \cdot 24$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Multiply $- 7$ by $24$ .

$16 + 23 y = - 168$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$16 + 23 y = - 168$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$16 + 23 y = - 168$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.3

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

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Step 6.3.3.1

Subtract $16$ from both sides of the equation.

$23 y = - 168 - 16$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.3.2

Subtract $16$ from $- 168$ .

$23 y = - 184$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$23 y = - 184$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.4

Divide each term in $23 y = - 184$ by $23$ and simplify.

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Step 6.3.4.1

Divide each term in $23 y = - 184$ by $23$ .

$\frac{23 y}{23} = \frac{- 184}{23}$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.4.2

Simplify the left side.

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Step 6.3.4.2.1

Cancel the common factor of $23$ .

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Step 6.3.4.2.1.1

Cancel the common factor.

$\frac{23 y}{23} = \frac{- 184}{23}$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 184}{23}$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$y = \frac{- 184}{23}$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$y = \frac{- 184}{23}$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.3.4.3

Simplify the right side.

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Step 6.3.4.3.1

Divide $- 184$ by $23$ .

$y = - 8$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$y = - 8$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$y = - 8$

$x = - \frac{4}{7} + \frac{3 y}{7}$

$y = - 8$

$x = - \frac{4}{7} + \frac{3 y}{7}$

Step 6.4

Replace all occurrences of $y$ with $- 8$ in each equation.

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Step 6.4.1

Replace all occurrences of $y$ in $x = - \frac{4}{7} + \frac{3 y}{7}$ with $- 8$ .

$x = - \frac{4}{7} + \frac{3 (- 8)}{7}$

$y = - 8$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $- \frac{4}{7} + \frac{3 (- 8)}{7}$ .

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Step 6.4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 4 + 3 (- 8)}{7}$

$y = - 8$

Step 6.4.2.1.2

Simplify the expression.

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Step 6.4.2.1.2.1

Multiply $3$ by $- 8$ .

$x = \frac{- 4 - 24}{7}$

$y = - 8$

Step 6.4.2.1.2.2

Subtract $24$ from $- 4$ .

$x = \frac{- 28}{7}$

$y = - 8$

Step 6.4.2.1.2.3

Divide $- 28$ by $7$ .

$x = - 4$

$y = - 8$

$x = - 4$

$y = - 8$

$x = - 4$

$y = - 8$

$x = - 4$

$y = - 8$

$x = - 4$

$y = - 8$

Step 6.5

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

$(- 4, - 8)$

Step 7

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

$m_{1} = \frac{7}{3}$

$m_{2} = \frac{4}{5}$

$(- 4, - 8)$

Step 8