Finite Math Examples

$(- a + 1, b - 1)$ , $(a + 1, - b)$

Step 1

Find the slope of the line between $(- a + 1, b - 1)$ and $(a + 1, - b)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- b - (b - 1)}{a + 1 - (- a + 1)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Apply the distributive property.

$m = \frac{- b - b + 1}{a + 1 - (- a + 1)}$

Step 1.4.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{- b - b + 1}{a + 1 - (- a + 1)}$

Step 1.4.1.3

Subtract $b$ from $- b$ .

$m = \frac{- 2 b + 1}{a + 1 - (- a + 1)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Apply the distributive property.

$m = \frac{- 2 b + 1}{a + 1 + a - 1 \cdot 1}$

Step 1.4.2.2

Multiply $- - a$ .

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Step 1.4.2.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 2 b + 1}{a + 1 + 1 a - 1 \cdot 1}$

Step 1.4.2.2.2

Multiply $a$ by $1$ .

$m = \frac{- 2 b + 1}{a + 1 + a - 1 \cdot 1}$

Step 1.4.2.3

Multiply $- 1$ by $1$ .

$m = \frac{- 2 b + 1}{a + 1 + a - 1}$

Step 1.4.2.4

Add $a$ and $a$ .

$m = \frac{- 2 b + 1}{2 a + 1 - 1}$

Step 1.4.2.5

Subtract $1$ from $1$ .

$m = \frac{- 2 b + 1}{2 a + 0}$

Step 1.4.2.6

Add $2 a$ and $0$ .

$m = \frac{- 2 b + 1}{2 a}$

Step 1.4.3

Simplify with factoring out.

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Step 1.4.3.1

Factor $- 1$ out of $- 2 b$ .

$m = \frac{- (2 b) + 1}{2 a}$

Step 1.4.3.2

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

$m = \frac{- (2 b) - 1 \cdot - 1}{2 a}$

Step 1.4.3.3

Factor $- 1$ out of $- (2 b) - 1 (- 1)$ .

$m = \frac{- (2 b - 1)}{2 a}$

Step 1.4.3.4

Simplify the expression.

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Step 1.4.3.4.1

Rewrite $- (2 b - 1)$ as $- 1 (2 b - 1)$ .

$m = \frac{- 1 (2 b - 1)}{2 a}$

Step 1.4.3.4.2

Move the negative in front of the fraction.

$m = - \frac{2 b - 1}{2 a}$

Step 2

Use the slope $- \frac{2 b - 1}{2 a}$ and a given point $(- a + 1, b - 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

Simplify the equation and keep it in point-slope form.

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

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{2 b - 1}{2 a} \cdot (x + a - 1)$ .

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Step 4.1.1

Rewrite.

$y - b + 1 = 0 + 0 - \frac{2 b - 1}{2 a} \cdot (x + a - 1)$

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Simplify.

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Step 4.1.4.1

Combine $x$ and $\frac{2 b - 1}{2 a}$ .

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

Step 4.1.4.2

Cancel the common factor of $a$ .

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Step 4.1.4.2.1

Move the leading negative in $- \frac{2 b - 1}{2 a}$ into the numerator.

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

Step 4.1.4.2.2

Factor $a$ out of $2 a$ .

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

Step 4.1.4.2.3

Cancel the common factor.

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

Step 4.1.4.2.4

Rewrite the expression.

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

Step 4.1.4.3

Multiply $- \frac{2 b - 1}{2 a} \cdot - 1$ .

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Step 4.1.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.4.3.2

Multiply $\frac{2 b - 1}{2 a}$ by $1$ .

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

Step 4.1.5

Move the negative in front of the fraction.

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

Step 4.1.6

To write $- \frac{2 b - 1}{2}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 4.1.7

Simplify terms.

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Step 4.1.7.1

Multiply $\frac{2 b - 1}{2}$ by $\frac{a}{a}$ .

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

Step 4.1.7.2

Combine the numerators over the common denominator.

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

Step 4.1.8

Simplify the numerator.

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Step 4.1.8.1

Factor $2 b - 1$ out of $- x (2 b - 1) - (2 b - 1) a$ .

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Step 4.1.8.1.1

Factor $2 b - 1$ out of $- x (2 b - 1)$ .

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

Step 4.1.8.1.2

Factor $2 b - 1$ out of $- (2 b - 1) a$ .

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

Step 4.1.8.1.3

Factor $2 b - 1$ out of $(2 b - 1) (- x) + (2 b - 1) (- 1 a)$ .

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

Step 4.1.8.2

Rewrite $- 1 a$ as $- a$ .

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

Step 4.1.9

Combine the numerators over the common denominator.

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

Step 4.1.10

Simplify the numerator.

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Step 4.1.10.1

Expand $(2 b - 1) (- x - a)$ using the FOIL Method.

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Step 4.1.10.1.1

Apply the distributive property.

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

Step 4.1.10.1.2

Apply the distributive property.

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

Step 4.1.10.1.3

Apply the distributive property.

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

Step 4.1.10.2

Simplify each term.

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Step 4.1.10.2.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.10.2.2

Multiply $2$ by $- 1$ .

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

Step 4.1.10.2.3

Rewrite using the commutative property of multiplication.

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

Step 4.1.10.2.4

Multiply $2$ by $- 1$ .

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

Step 4.1.10.2.5

Multiply $- 1 (- x)$ .

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Step 4.1.10.2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.10.2.5.2

Multiply $x$ by $1$ .

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

Step 4.1.10.2.6

Multiply $- 1 (- a)$ .

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Step 4.1.10.2.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.10.2.6.2

Multiply $a$ by $1$ .

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

Step 4.1.11

Simplify with factoring out.

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Step 4.1.11.1

Factor $- 1$ out of $- 2 b x$ .

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

Step 4.1.11.2

Factor $- 1$ out of $- 2 b a$ .

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

Step 4.1.11.3

Factor $- 1$ out of $- (2 b x) - (2 b a)$ .

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

Step 4.1.11.4

Factor $- 1$ out of $x$ .

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

Step 4.1.11.5

Factor $- 1$ out of $- (2 b x + 2 b a) - 1 (- x)$ .

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

Step 4.1.11.6

Factor $- 1$ out of $a$ .

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

Step 4.1.11.7

Factor $- 1$ out of $- (2 b x + 2 b a - x) - 1 (- a)$ .

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

Step 4.1.11.8

Factor $- 1$ out of $2 b$ .

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

Step 4.1.11.9

Factor $- 1$ out of $- (2 b x + 2 b a - x - a) - (- 2 b)$ .

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

Step 4.1.11.10

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

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

Step 4.1.11.11

Factor $- 1$ out of $- (2 b x + 2 b a - x - a - 2 b) - 1 (1)$ .

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

Step 4.1.11.12

Simplify the expression.

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Step 4.1.11.12.1

Rewrite $- (2 b x + 2 b a - x - a - 2 b + 1)$ as $- 1 (2 b x + 2 b a - x - a - 2 b + 1)$ .

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

Step 4.1.11.12.2

Move the negative in front of the fraction.

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

Step 4.2

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

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Step 4.2.1

Add $b$ to both sides of the equation.

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

Step 4.2.2

Subtract $1$ from both sides of the equation.

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Split the fraction $\frac{2 b x + 2 b a - x - a - 2 b + 1}{2 a}$ into two fractions.

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

Step 4.2.3.2

Simplify each term.

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Step 4.2.3.2.1

Split the fraction $\frac{2 b x + 2 b a - x - a - 2 b}{2 a}$ into two fractions.

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

Step 4.2.3.2.2

Simplify each term.

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Step 4.2.3.2.2.1

Simplify the numerator.

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Step 4.2.3.2.2.1.1

Factor out the greatest common factor from each group.

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Step 4.2.3.2.2.1.1.1

Group the first two terms and the last two terms.

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

Step 4.2.3.2.2.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.2.3.2.2.1.2

Factor the polynomial by factoring out the greatest common factor, $x + a$ .

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

Step 4.2.3.2.2.2

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

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Step 4.2.3.2.2.2.1

Factor $2$ out of $- 2 b$ .

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

Step 4.2.3.2.2.2.2

Cancel the common factors.

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Step 4.2.3.2.2.2.2.1

Factor $2$ out of $2 a$ .

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

Step 4.2.3.2.2.2.2.2

Cancel the common factor.

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

Step 4.2.3.2.2.2.2.3

Rewrite the expression.

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

Step 4.2.3.2.2.3

Move the negative in front of the fraction.

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

Step 4.2.3.3

Apply the distributive property.

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

Step 4.2.3.4

Multiply $- - \frac{b}{a}$ .

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Step 4.2.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3.4.2

Multiply $\frac{b}{a}$ by $1$ .

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

Step 5

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 6