Finite Math Examples

$x = y + 1$ , $5 x + 2 y = - 23$

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

Rewrite the equation as $y + 1 = x$ .

$y + 1 = x$

Step 1.3

Subtract $1$ from both sides of the equation.

$y = x - 1$

Step 2

Using the slope-intercept form, the slope is $1$ .

$m_{1} = 1$

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 $5 x$ from both sides of the equation.

$2 y = - 23 - 5 x$

Step 3.3

Divide each term in $2 y = - 23 - 5 x$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 y = - 23 - 5 x$ by $2$ .

$\frac{2 y}{2} = \frac{- 23}{2} + \frac{- 5 x}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 23}{2} + \frac{- 5 x}{2}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 23}{2} + \frac{- 5 x}{2}$

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.

$y = - \frac{23}{2} + \frac{- 5 x}{2}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{23}{2} - \frac{5 x}{2}$

Step 3.4

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

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Step 3.4.1

Reorder $- \frac{23}{2}$ and $- \frac{5 x}{2}$ .

$y = - \frac{5 x}{2} - \frac{23}{2}$

Step 3.4.2

Reorder terms.

$y = - (\frac{5}{2} x) - \frac{23}{2}$

Step 3.4.3

Remove parentheses.

$y = - \frac{5}{2} x - \frac{23}{2}$

Step 4

Using the slope-intercept form, the slope is $- \frac{5}{2}$ .

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

Step 5

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

$x = y + 1, 5 x + 2 y = - 23$

Step 6

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

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Step 6.1

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

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Step 6.1.1

Replace all occurrences of $x$ in $5 x + 2 y = - 23$ with $y + 1$ .

$5 (y + 1) + 2 y = - 23$

$x = y + 1$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Simplify $5 (y + 1) + 2 y$ .

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Step 6.1.2.1.1

Simplify each term.

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Step 6.1.2.1.1.1

Apply the distributive property.

$5 y + 5 \cdot 1 + 2 y = - 23$

$x = y + 1$

Step 6.1.2.1.1.2

Multiply $5$ by $1$ .

$5 y + 5 + 2 y = - 23$

$x = y + 1$

$5 y + 5 + 2 y = - 23$

$x = y + 1$

Step 6.1.2.1.2

Add $5 y$ and $2 y$ .

$7 y + 5 = - 23$

$x = y + 1$

$7 y + 5 = - 23$

$x = y + 1$

$7 y + 5 = - 23$

$x = y + 1$

$7 y + 5 = - 23$

$x = y + 1$

Step 6.2

Solve for $y$ in $7 y + 5 = - 23$ .

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Step 6.2.1

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

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Step 6.2.1.1

Subtract $5$ from both sides of the equation.

$7 y = - 23 - 5$

$x = y + 1$

Step 6.2.1.2

Subtract $5$ from $- 23$ .

$7 y = - 28$

$x = y + 1$

$7 y = - 28$

$x = y + 1$

Step 6.2.2

Divide each term in $7 y = - 28$ by $7$ and simplify.

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Step 6.2.2.1

Divide each term in $7 y = - 28$ by $7$ .

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

$x = y + 1$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $7$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

$x = y + 1$

Step 6.2.2.2.1.2

Divide $y$ by $1$ .

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

$x = y + 1$

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

$x = y + 1$

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

$x = y + 1$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Divide $- 28$ by $7$ .

$y = - 4$

$x = y + 1$

$y = - 4$

$x = y + 1$

$y = - 4$

$x = y + 1$

$y = - 4$

$x = y + 1$

Step 6.3

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 6.3.1

Replace all occurrences of $y$ in $x = y + 1$ with $- 4$ .

$x = (- 4) + 1$

$y = - 4$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Add $- 4$ and $1$ .

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

Step 6.4

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

$(- 3, - 4)$

Step 7

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

$m_{1} = 1$

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

$(- 3, - 4)$

Step 8