Finite Math Examples

$(5, 6)$ , $x + 6 y = 5$

Step 1

Solve the equation for $y$ in terms of $x$ .

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Step 1.1

Subtract $x$ from both sides of the equation.

$6 y = 5 - x$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

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

Step 2

The Intermediate Value Theorem states that, if $f$ is a real-valued continuous function on the interval $[a, b]$ , and $u$ is a number between $f (a)$ and $f (b)$ , then there is a $c$ contained in the interval $[a, b]$ such that $f (c) = u$ .

$u = f (c) = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Calculate $f (a) = f (5) = \frac{5}{6} - \frac{5}{6}$ .

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Step 4.1

Combine the numerators over the common denominator.

$f (5) = \frac{5 - 5}{6}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Subtract $5$ from $5$ .

$f (5) = \frac{0}{6}$

Step 4.2.2

Divide $0$ by $6$ .

$f (5) = 0$

Step 5

Calculate $f (b) = f (6) = \frac{5}{6} - \frac{6}{6}$ .

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Step 5.1

Combine the numerators over the common denominator.

$f (6) = \frac{5 - 6}{6}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Subtract $6$ from $5$ .

$f (6) = \frac{- 1}{6}$

Step 5.2.2

Move the negative in front of the fraction.

$f (6) = - \frac{1}{6}$

Step 6

Since $0$ is on the interval $[- \frac{1}{6}, 0]$ , solve the equation for $x$ at the root by setting $y$ to $0$ in $y = \frac{5}{6} - \frac{x}{6}$ .

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Step 6.1

Rewrite the equation as $\frac{5}{6} - \frac{x}{6} = 0$ .

$\frac{5}{6} - \frac{x}{6} = 0$

Step 6.2

Subtract $\frac{5}{6}$ from both sides of the equation.

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

Step 6.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x = - 5$

Step 6.4

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 6.4.1

Divide each term in $- x = - 5$ by $- 1$ .

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

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Dividing two negative values results in a positive value.

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

Step 6.4.2.2

Divide $x$ by $1$ .

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

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 7

The Intermediate Value Theorem states that there is a root $f (c) = 0$ on the interval $[- \frac{1}{6}, 0]$ because $f$ is a continuous function on $[5, 6]$ .

The roots on the interval $[5, 6]$ are located at .

Step 8