Finite Math Examples

$3 x - 4 y = 16$ , $6 x + 8 y = - 2$

Step 1

Rewrite in slope-intercept form.

Tap for more steps...

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.

$- 4 y = 16 - 3 x$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 1.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.1

Simplify each term.

Tap for more steps...

Step 1.3.3.1.1

Divide $16$ by $- 4$ .

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

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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.

Tap for more steps...

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 $6 x$ from both sides of the equation.

$8 y = - 2 - 6 x$

Step 3.3

Divide each term in $8 y = - 2 - 6 x$ by $8$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $8 y = - 2 - 6 x$ by $8$ .

$\frac{8 y}{8} = \frac{- 2}{8} + \frac{- 6 x}{8}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 2}{8} + \frac{- 6 x}{8}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{8} + \frac{- 6 x}{8}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.3.1.1

Cancel the common factor of $- 2$ and $8$ .

Tap for more steps...

Step 3.3.3.1.1.1

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1)}{8} + \frac{- 6 x}{8}$

Step 3.3.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.1.2.1

Factor $2$ out of $8$ .

$y = \frac{2 \cdot - 1}{2 \cdot 4} + \frac{- 6 x}{8}$

Step 3.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 1}{2 \cdot 4} + \frac{- 6 x}{8}$

Step 3.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{- 1}{4} + \frac{- 6 x}{8}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{4} + \frac{- 6 x}{8}$

Step 3.3.3.1.3

Cancel the common factor of $- 6$ and $8$ .

Tap for more steps...

Step 3.3.3.1.3.1

Factor $2$ out of $- 6 x$ .

$y = - \frac{1}{4} + \frac{2 (- 3 x)}{8}$

Step 3.3.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.3.2.1

Factor $2$ out of $8$ .

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

Step 3.3.3.1.3.2.2

Cancel the common factor.

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

Step 3.3.3.1.3.2.3

Rewrite the expression.

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

Step 3.3.3.1.4

Move the negative in front of the fraction.

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

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Reorder terms.

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

Step 3.4.3

Remove parentheses.

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

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.

$3 x - 4 y = 16, 6 x + 8 y = - 2$

Step 6

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

Tap for more steps...

Step 6.1

Solve for $x$ in $3 x - 4 y = 16$ .

Tap for more steps...

Step 6.1.1

Add $4 y$ to both sides of the equation.

$3 x = 16 + 4 y$

$6 x + 8 y = - 2$

Step 6.1.2

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

Tap for more steps...

Step 6.1.2.1

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

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

$6 x + 8 y = - 2$

Step 6.1.2.2

Simplify the left side.

Tap for more steps...

Step 6.1.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.1.2.2.1.1

Cancel the common factor.

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

$6 x + 8 y = - 2$

Step 6.1.2.2.1.2

Divide $x$ by $1$ .

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

$6 x + 8 y = - 2$

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

$6 x + 8 y = - 2$

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

$6 x + 8 y = - 2$

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

$6 x + 8 y = - 2$

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

$6 x + 8 y = - 2$

Step 6.2

Replace all occurrences of $x$ with $\frac{16}{3} + \frac{4 y}{3}$ in each equation.

Tap for more steps...

Step 6.2.1

Replace all occurrences of $x$ in $6 x + 8 y = - 2$ with $\frac{16}{3} + \frac{4 y}{3}$ .

$6 (\frac{16}{3} + \frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Simplify $6 (\frac{16}{3} + \frac{4 y}{3}) + 8 y$ .

Tap for more steps...

Step 6.2.2.1.1

Simplify each term.

Tap for more steps...

Step 6.2.2.1.1.1

Apply the distributive property.

$6 (\frac{16}{3}) + 6 (\frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.2.1.1.2.1

Factor $3$ out of $6$ .

$3 (2) (\frac{16}{3}) + 6 (\frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2.1.1.2.2

Cancel the common factor.

$3 \cdot (2 (\frac{16}{3})) + 6 (\frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2.1.1.2.3

Rewrite the expression.

$2 \cdot 16 + 6 (\frac{4 y}{3}) + 8 y = - 2$

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

$2 \cdot 16 + 6 (\frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2.1.1.3

Multiply $2$ by $16$ .

$32 + 6 (\frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2.1.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.2.1.1.4.1

Factor $3$ out of $6$ .

$32 + 3 (2) (\frac{4 y}{3}) + 8 y = - 2$

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

Step 6.2.2.1.1.4.2

Cancel the common factor.

$32 + 3 \cdot (2 (\frac{4 y}{3})) + 8 y = - 2$

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

Step 6.2.2.1.1.4.3

Rewrite the expression.

$32 + 2 (4 y) + 8 y = - 2$

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

$32 + 2 (4 y) + 8 y = - 2$

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

Step 6.2.2.1.1.5

Multiply $4$ by $2$ .

$32 + 8 y + 8 y = - 2$

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

$32 + 8 y + 8 y = - 2$

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

Step 6.2.2.1.2

Add $8 y$ and $8 y$ .

$32 + 16 y = - 2$

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

$32 + 16 y = - 2$

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

$32 + 16 y = - 2$

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

$32 + 16 y = - 2$

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

Step 6.3

Solve for $y$ in $32 + 16 y = - 2$ .

Tap for more steps...

Step 6.3.1

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

Tap for more steps...

Step 6.3.1.1

Subtract $32$ from both sides of the equation.

$16 y = - 2 - 32$

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

Step 6.3.1.2

Subtract $32$ from $- 2$ .

$16 y = - 34$

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

$16 y = - 34$

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

Step 6.3.2

Divide each term in $16 y = - 34$ by $16$ and simplify.

Tap for more steps...

Step 6.3.2.1

Divide each term in $16 y = - 34$ by $16$ .

$\frac{16 y}{16} = \frac{- 34}{16}$

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

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

Cancel the common factor of $16$ .

Tap for more steps...

Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{16 y}{16} = \frac{- 34}{16}$

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

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 34}{16}$

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

$y = \frac{- 34}{16}$

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

$y = \frac{- 34}{16}$

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

Step 6.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.3.1

Cancel the common factor of $- 34$ and $16$ .

Tap for more steps...

Step 6.3.2.3.1.1

Factor $2$ out of $- 34$ .

$y = \frac{2 (- 17)}{16}$

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

Step 6.3.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.3.1.2.1

Factor $2$ out of $16$ .

$y = \frac{2 \cdot - 17}{2 \cdot 8}$

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

Step 6.3.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 17}{2 \cdot 8}$

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

Step 6.3.2.3.1.2.3

Rewrite the expression.

$y = \frac{- 17}{8}$

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

$y = \frac{- 17}{8}$

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

$y = \frac{- 17}{8}$

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

Step 6.3.2.3.2

Move the negative in front of the fraction.

$y = - \frac{17}{8}$

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

$y = - \frac{17}{8}$

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

$y = - \frac{17}{8}$

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

$y = - \frac{17}{8}$

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

Step 6.4

Replace all occurrences of $y$ with $- \frac{17}{8}$ in each equation.

Tap for more steps...

Step 6.4.1

Replace all occurrences of $y$ in $x = \frac{16}{3} + \frac{4 y}{3}$ with $- \frac{17}{8}$ .

$x = \frac{16}{3} + \frac{4 (- \frac{17}{8})}{3}$

$y = - \frac{17}{8}$

Step 6.4.2

Simplify the right side.

Tap for more steps...

Step 6.4.2.1

Simplify $\frac{16}{3} + \frac{4 (- \frac{17}{8})}{3}$ .

Tap for more steps...

Step 6.4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{16 + 4 (- \frac{17}{8})}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.2

Simplify each term.

Tap for more steps...

Step 6.4.2.1.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.4.2.1.2.1.1

Move the leading negative in $- \frac{17}{8}$ into the numerator.

$x = \frac{16 + 4 (\frac{- 17}{8})}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.2.1.2

Factor $4$ out of $8$ .

$x = \frac{16 + 4 (\frac{- 17}{4 (2)})}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.2.1.3

Cancel the common factor.

$x = \frac{16 + 4 (\frac{- 17}{4 \cdot 2})}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.2.1.4

Rewrite the expression.

$x = \frac{16 + \frac{- 17}{2}}{3}$

$y = - \frac{17}{8}$

$x = \frac{16 + \frac{- 17}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{16 - \frac{17}{2}}{3}$

$y = - \frac{17}{8}$

$x = \frac{16 - \frac{17}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.3

To write $16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{16 \cdot \frac{2}{2} - \frac{17}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.4

Combine $16$ and $\frac{2}{2}$ .

$x = \frac{\frac{16 \cdot 2}{2} - \frac{17}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{16 \cdot 2 - 17}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.6

Simplify the numerator.

Tap for more steps...

Step 6.4.2.1.6.1

Multiply $16$ by $2$ .

$x = \frac{\frac{32 - 17}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.6.2

Subtract $17$ from $32$ .

$x = \frac{\frac{15}{2}}{3}$

$y = - \frac{17}{8}$

$x = \frac{\frac{15}{2}}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{15}{2} \cdot \frac{1}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.8

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.4.2.1.8.1

Factor $3$ out of $15$ .

$x = \frac{3 (5)}{2} \cdot \frac{1}{3}$

$y = - \frac{17}{8}$

Step 6.4.2.1.8.2

Cancel the common factor.

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

$y = - \frac{17}{8}$

Step 6.4.2.1.8.3

Rewrite the expression.

$x = \frac{5}{2}$

$y = - \frac{17}{8}$

$x = \frac{5}{2}$

$y = - \frac{17}{8}$

$x = \frac{5}{2}$

$y = - \frac{17}{8}$

$x = \frac{5}{2}$

$y = - \frac{17}{8}$

$x = \frac{5}{2}$

$y = - \frac{17}{8}$

Step 6.5

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

$(\frac{5}{2}, - \frac{17}{8})$

Step 7

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

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

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

$(\frac{5}{2}, - \frac{17}{8})$

Step 8