Finite Math Examples

$x + 6 y = - 5$ , $4 x + 2 y = 2$

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 $x$ from both sides of the equation.

$6 y = - 5 - x$

Step 1.3

Divide each term in $6 y = - 5 - x$ by $6$ and simplify.

Tap for more steps...

Step 1.3.1

Divide each term in $6 y = - 5 - x$ by $6$ .

$\frac{6 y}{6} = \frac{- 5}{6} + \frac{- x}{6}$

Step 1.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 1.3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 5}{6} + \frac{- x}{6}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{6} + \frac{- x}{6}$

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

Move the negative in front of the fraction.

$y = - \frac{5}{6} + \frac{- x}{6}$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{5}{6} - \frac{x}{6}$

Step 1.4

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

Tap for more steps...

Step 1.4.1

Reorder $- \frac{5}{6}$ and $- \frac{x}{6}$ .

$y = - \frac{x}{6} - \frac{5}{6}$

Step 1.4.2

Reorder terms.

$y = - (\frac{1}{6} x) - \frac{5}{6}$

Step 1.4.3

Remove parentheses.

$y = - \frac{1}{6} x - \frac{5}{6}$

Step 2

Using the slope-intercept form, the slope is $- \frac{1}{6}$ .

$m_{1} = - \frac{1}{6}$

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 $4 x$ from both sides of the equation.

$2 y = 2 - 4 x$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

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

Divide $2$ by $2$ .

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

Step 3.3.3.1.2

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

Tap for more steps...

Step 3.3.3.1.2.1

Factor $2$ out of $- 4 x$ .

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

Step 3.3.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.2.1

Factor $2$ out of $2$ .

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

Step 3.3.3.1.2.2.2

Cancel the common factor.

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

Step 3.3.3.1.2.2.3

Rewrite the expression.

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

Step 3.3.3.1.2.2.4

Divide $- 2 x$ by $1$ .

$y = 1 - 2 x$

Step 3.4

Reorder $1$ and $- 2 x$ .

$y = - 2 x + 1$

Step 4

Using the slope-intercept form, the slope is $- 2$ .

$m_{2} = - 2$

Step 5

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

$x + 6 y = - 5, 4 x + 2 y = 2$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $6 y$ from both sides of the equation.

$x = - 5 - 6 y$

$4 x + 2 y = 2$

Step 6.2

Replace all occurrences of $x$ with $- 5 - 6 y$ in each equation.

Tap for more steps...

Step 6.2.1

Replace all occurrences of $x$ in $4 x + 2 y = 2$ with $- 5 - 6 y$ .

$4 (- 5 - 6 y) + 2 y = 2$

$x = - 5 - 6 y$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Simplify $4 (- 5 - 6 y) + 2 y$ .

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.

$4 \cdot - 5 + 4 (- 6 y) + 2 y = 2$

$x = - 5 - 6 y$

Step 6.2.2.1.1.2

Multiply $4$ by $- 5$ .

$- 20 + 4 (- 6 y) + 2 y = 2$

$x = - 5 - 6 y$

Step 6.2.2.1.1.3

Multiply $- 6$ by $4$ .

$- 20 - 24 y + 2 y = 2$

$x = - 5 - 6 y$

$- 20 - 24 y + 2 y = 2$

$x = - 5 - 6 y$

Step 6.2.2.1.2

Add $- 24 y$ and $2 y$ .

$- 20 - 22 y = 2$

$x = - 5 - 6 y$

$- 20 - 22 y = 2$

$x = - 5 - 6 y$

$- 20 - 22 y = 2$

$x = - 5 - 6 y$

$- 20 - 22 y = 2$

$x = - 5 - 6 y$

Step 6.3

Solve for $y$ in $- 20 - 22 y = 2$ .

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

Add $20$ to both sides of the equation.

$- 22 y = 2 + 20$

$x = - 5 - 6 y$

Step 6.3.1.2

Add $2$ and $20$ .

$- 22 y = 22$

$x = - 5 - 6 y$

$- 22 y = 22$

$x = - 5 - 6 y$

Step 6.3.2

Divide each term in $- 22 y = 22$ by $- 22$ and simplify.

Tap for more steps...

Step 6.3.2.1

Divide each term in $- 22 y = 22$ by $- 22$ .

$\frac{- 22 y}{- 22} = \frac{22}{- 22}$

$x = - 5 - 6 y$

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

Cancel the common factor of $- 22$ .

Tap for more steps...

Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 22 y}{- 22} = \frac{22}{- 22}$

$x = - 5 - 6 y$

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{22}{- 22}$

$x = - 5 - 6 y$

$y = \frac{22}{- 22}$

$x = - 5 - 6 y$

$y = \frac{22}{- 22}$

$x = - 5 - 6 y$

Step 6.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.3.1

Divide $22$ by $- 22$ .

$y = - 1$

$x = - 5 - 6 y$

$y = - 1$

$x = - 5 - 6 y$

$y = - 1$

$x = - 5 - 6 y$

$y = - 1$

$x = - 5 - 6 y$

Step 6.4

Replace all occurrences of $y$ with $- 1$ in each equation.

Tap for more steps...

Step 6.4.1

Replace all occurrences of $y$ in $x = - 5 - 6 y$ with $- 1$ .

$x = - 5 - 6 \cdot - 1$

$y = - 1$

Step 6.4.2

Simplify the right side.

Tap for more steps...

Step 6.4.2.1

Simplify $- 5 - 6 \cdot - 1$ .

Tap for more steps...

Step 6.4.2.1.1

Multiply $- 6$ by $- 1$ .

$x = - 5 + 6$

$y = - 1$

Step 6.4.2.1.2

Add $- 5$ and $6$ .

$x = 1$

$y = - 1$

$x = 1$

$y = - 1$

$x = 1$

$y = - 1$

$x = 1$

$y = - 1$

Step 6.5

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

$(1, - 1)$

Step 7

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

$m_{1} = - \frac{1}{6}$

$m_{2} = - 2$

$(1, - 1)$

Step 8