Finite Math Examples

$7 x + y = - 1$ , $- 14 x + 12 y = 2$

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 $7 x$ from both sides of the equation.

$y = - 1 - 7 x$

Step 1.3

Reorder $- 1$ and $- 7 x$ .

$y = - 7 x - 1$

Step 2

Using the slope-intercept form, the slope is $- 7$ .

$m_{1} = - 7$

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

Add $14 x$ to both sides of the equation.

$12 y = 2 + 14 x$

Step 3.3

Divide each term in $12 y = 2 + 14 x$ by $12$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $12 y = 2 + 14 x$ by $12$ .

$\frac{12 y}{12} = \frac{2}{12} + \frac{14 x}{12}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{2}{12} + \frac{14 x}{12}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{12} + \frac{14 x}{12}$

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 $12$ .

Tap for more steps...

Step 3.3.3.1.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{12} + \frac{14 x}{12}$

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 $12$ .

$y = \frac{2 \cdot 1}{2 \cdot 6} + \frac{14 x}{12}$

Step 3.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot 6} + \frac{14 x}{12}$

Step 3.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{6} + \frac{14 x}{12}$

Step 3.3.3.1.2

Cancel the common factor of $14$ and $12$ .

Tap for more steps...

Step 3.3.3.1.2.1

Factor $2$ out of $14 x$ .

$y = \frac{1}{6} + \frac{2 (7 x)}{12}$

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 $12$ .

$y = \frac{1}{6} + \frac{2 (7 x)}{2 (6)}$

Step 3.3.3.1.2.2.2

Cancel the common factor.

$y = \frac{1}{6} + \frac{2 (7 x)}{2 \cdot 6}$

Step 3.3.3.1.2.2.3

Rewrite the expression.

$y = \frac{1}{6} + \frac{7 x}{6}$

Step 3.4

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

Tap for more steps...

Step 3.4.1

Reorder $\frac{1}{6}$ and $\frac{7 x}{6}$ .

$y = \frac{7 x}{6} + \frac{1}{6}$

Step 3.4.2

Reorder terms.

$y = \frac{7}{6} x + \frac{1}{6}$

Step 4

Using the slope-intercept form, the slope is $\frac{7}{6}$ .

$m_{2} = \frac{7}{6}$

Step 5

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

$7 x + y = - 1, - 14 x + 12 y = 2$

Step 6

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

Tap for more steps...

Step 6.1

Subtract $7 x$ from both sides of the equation.

$y = - 1 - 7 x$

$- 14 x + 12 y = 2$

Step 6.2

Replace all occurrences of $y$ with $- 1 - 7 x$ in each equation.

Tap for more steps...

Step 6.2.1

Replace all occurrences of $y$ in $- 14 x + 12 y = 2$ with $- 1 - 7 x$ .

$- 14 x + 12 (- 1 - 7 x) = 2$

$y = - 1 - 7 x$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Simplify $- 14 x + 12 (- 1 - 7 x)$ .

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.

$- 14 x + 12 \cdot - 1 + 12 (- 7 x) = 2$

$y = - 1 - 7 x$

Step 6.2.2.1.1.2

Multiply $12$ by $- 1$ .

$- 14 x - 12 + 12 (- 7 x) = 2$

$y = - 1 - 7 x$

Step 6.2.2.1.1.3

Multiply $- 7$ by $12$ .

$- 14 x - 12 - 84 x = 2$

$y = - 1 - 7 x$

$- 14 x - 12 - 84 x = 2$

$y = - 1 - 7 x$

Step 6.2.2.1.2

Subtract $84 x$ from $- 14 x$ .

$- 98 x - 12 = 2$

$y = - 1 - 7 x$

$- 98 x - 12 = 2$

$y = - 1 - 7 x$

$- 98 x - 12 = 2$

$y = - 1 - 7 x$

$- 98 x - 12 = 2$

$y = - 1 - 7 x$

Step 6.3

Solve for $x$ in $- 98 x - 12 = 2$ .

Tap for more steps...

Step 6.3.1

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

Tap for more steps...

Step 6.3.1.1

Add $12$ to both sides of the equation.

$- 98 x = 2 + 12$

$y = - 1 - 7 x$

Step 6.3.1.2

Add $2$ and $12$ .

$- 98 x = 14$

$y = - 1 - 7 x$

$- 98 x = 14$

$y = - 1 - 7 x$

Step 6.3.2

Divide each term in $- 98 x = 14$ by $- 98$ and simplify.

Tap for more steps...

Step 6.3.2.1

Divide each term in $- 98 x = 14$ by $- 98$ .

$\frac{- 98 x}{- 98} = \frac{14}{- 98}$

$y = - 1 - 7 x$

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

Cancel the common factor of $- 98$ .

Tap for more steps...

Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 98 x}{- 98} = \frac{14}{- 98}$

$y = - 1 - 7 x$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{- 98}$

$y = - 1 - 7 x$

$x = \frac{14}{- 98}$

$y = - 1 - 7 x$

$x = \frac{14}{- 98}$

$y = - 1 - 7 x$

Step 6.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.3.1

Cancel the common factor of $14$ and $- 98$ .

Tap for more steps...

Step 6.3.2.3.1.1

Factor $14$ out of $14$ .

$x = \frac{14 (1)}{- 98}$

$y = - 1 - 7 x$

Step 6.3.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.3.1.2.1

Factor $14$ out of $- 98$ .

$x = \frac{14 \cdot 1}{14 \cdot - 7}$

$y = - 1 - 7 x$

Step 6.3.2.3.1.2.2

Cancel the common factor.

$x = \frac{14 \cdot 1}{14 \cdot - 7}$

$y = - 1 - 7 x$

Step 6.3.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{- 7}$

$y = - 1 - 7 x$

$x = \frac{1}{- 7}$

$y = - 1 - 7 x$

$x = \frac{1}{- 7}$

$y = - 1 - 7 x$

Step 6.3.2.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{7}$

$y = - 1 - 7 x$

$x = - \frac{1}{7}$

$y = - 1 - 7 x$

$x = - \frac{1}{7}$

$y = - 1 - 7 x$

$x = - \frac{1}{7}$

$y = - 1 - 7 x$

Step 6.4

Replace all occurrences of $x$ with $- \frac{1}{7}$ in each equation.

Tap for more steps...

Step 6.4.1

Replace all occurrences of $x$ in $y = - 1 - 7 x$ with $- \frac{1}{7}$ .

$y = - 1 - 7 (- \frac{1}{7})$

$x = - \frac{1}{7}$

Step 6.4.2

Simplify the right side.

Tap for more steps...

Step 6.4.2.1

Simplify $- 1 - 7 (- \frac{1}{7})$ .

Tap for more steps...

Step 6.4.2.1.1

Simplify each term.

Tap for more steps...

Step 6.4.2.1.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 6.4.2.1.1.1.1

Move the leading negative in $- \frac{1}{7}$ into the numerator.

$y = - 1 - 7 (\frac{- 1}{7})$

$x = - \frac{1}{7}$

Step 6.4.2.1.1.1.2

Factor $7$ out of $- 7$ .

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

$x = - \frac{1}{7}$

Step 6.4.2.1.1.1.3

Cancel the common factor.

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

$x = - \frac{1}{7}$

Step 6.4.2.1.1.1.4

Rewrite the expression.

$y = - 1 - 1 \cdot - 1$

$x = - \frac{1}{7}$

$y = - 1 - 1 \cdot - 1$

$x = - \frac{1}{7}$

Step 6.4.2.1.1.2

Multiply $- 1$ by $- 1$ .

$y = - 1 + 1$

$x = - \frac{1}{7}$

$y = - 1 + 1$

$x = - \frac{1}{7}$

Step 6.4.2.1.2

Add $- 1$ and $1$ .

$y = 0$

$x = - \frac{1}{7}$

$y = 0$

$x = - \frac{1}{7}$

$y = 0$

$x = - \frac{1}{7}$

$y = 0$

$x = - \frac{1}{7}$

Step 6.5

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

$(- \frac{1}{7}, 0)$

Step 7

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

$m_{1} = - 7$

$m_{2} = \frac{7}{6}$

$(- \frac{1}{7}, 0)$

Step 8