Pre-Algebra Examples

$\frac{x - 2}{5} + \frac{{(y - 5)}^{2}}{5} = 1$

Step 1

Combine the numerators over the common denominator.

$\frac{x - 2 + {(y - 5)}^{2}}{5} = 1$

Step 2

Multiply both sides of the equation by $5$ .

$5 \frac{x - 2 + {(y - 5)}^{2}}{5} = 5 \cdot 1$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $5$ .

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Step 3.1.1.1

Cancel the common factor.

$5 \frac{x - 2 + {(y - 5)}^{2}}{5} = 5 \cdot 1$

Step 3.1.1.2

Rewrite the expression.

$x - 2 + {(y - 5)}^{2} = 5 \cdot 1$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $5$ by $1$ .

$x - 2 + {(y - 5)}^{2} = 5$

Step 4

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

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Step 4.1

Add $2$ to both sides of the equation.

$x + {(y - 5)}^{2} = 5 + 2$

Step 4.2

Subtract ${(y - 5)}^{2}$ from both sides of the equation.

$x = 5 + 2 - {(y - 5)}^{2}$

Step 4.3

Add $5$ and $2$ .

$x = 7 - {(y - 5)}^{2}$

Step 5

Find the properties of the given parabola.

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Step 5.1

Reorder $7$ and $- {(y - 5)}^{2}$ .

$x = - {(y - 5)}^{2} + 7$

Step 5.2

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 1$

$h = 7$

$k = 5$

Step 5.3

Since the value of $a$ is negative, the parabola opens left.

Opens Left

Step 5.4

Find the vertex $(h, k)$ .

$(7, 5)$

Step 5.5

Find $p$ , the distance from the vertex to the focus.

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Step 5.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 1}$

Step 5.5.3

Cancel the common factor of $1$ and $- 1$ .

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Step 5.5.3.1

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

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Step 5.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5.6

Find the focus.

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Step 5.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 5.6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{27}{4}, 5)$

Step 5.7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$y = 5$

Step 5.8

Find the directrix.

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Step 5.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 5.8.2

Substitute the known values of $p$ and $h$ into the formula and simplify.

$x = \frac{29}{4}$

Step 5.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Left

Vertex: $(7, 5)$

Focus: $(\frac{27}{4}, 5)$

Axis of Symmetry: $y = 5$

Directrix: $x = \frac{29}{4}$

Direction: Opens Left

Vertex: $(7, 5)$

Focus: $(\frac{27}{4}, 5)$

Axis of Symmetry: $y = 5$

Directrix: $x = \frac{29}{4}$

Step 6

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 6.1

Substitute the $x$ value $5$ into $f (x) = \sqrt{- x + 7} + 5$ . In this case, the point is $(5, 6.41421356)$ .

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Step 6.1.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \sqrt{- (5) + 7} + 5$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Multiply $- 1$ by $5$ .

$f (5) = \sqrt{- 5 + 7} + 5$

Step 6.1.2.1.2

Add $- 5$ and $7$ .

$f (5) = \sqrt{2} + 5$

Step 6.1.2.2

The final answer is $\sqrt{2} + 5$ .

$y = \sqrt{2} + 5$

Step 6.1.3

Convert $\sqrt{2} + 5$ to decimal.

$= 6.41421356$

Step 6.2

Substitute the $x$ value $5$ into $f (x) = - \sqrt{- x + 7} + 5$ . In this case, the point is $(5, 3.58578643)$ .

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Step 6.2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \sqrt{- (5) + 7} + 5$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $- 1$ by $5$ .

$f (5) = - \sqrt{- 5 + 7} + 5$

Step 6.2.2.1.2

Add $- 5$ and $7$ .

$f (5) = - \sqrt{2} + 5$

Step 6.2.2.2

The final answer is $- \sqrt{2} + 5$ .

$y = - \sqrt{2} + 5$

Step 6.2.3

Convert $- \sqrt{2} + 5$ to decimal.

$= 3.58578643$

Step 6.3

Substitute the $x$ value $6$ into $f (x) = \sqrt{- x + 7} + 5$ . In this case, the point is $(6, 6)$ .

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Step 6.3.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \sqrt{- (6) + 7} + 5$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Multiply $- 1$ by $6$ .

$f (6) = \sqrt{- 6 + 7} + 5$

Step 6.3.2.1.2

Add $- 6$ and $7$ .

$f (6) = \sqrt{1} + 5$

Step 6.3.2.1.3

Any root of $1$ is $1$ .

$f (6) = 1 + 5$

Step 6.3.2.2

Add $1$ and $5$ .

$f (6) = 6$

Step 6.3.2.3

The final answer is $6$ .

$y = 6$

Step 6.3.3

Convert $6$ to decimal.

$= 6$

Step 6.4

Substitute the $x$ value $6$ into $f (x) = - \sqrt{- x + 7} + 5$ . In this case, the point is $(6, 4)$ .

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Step 6.4.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = - \sqrt{- (6) + 7} + 5$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Multiply $- 1$ by $6$ .

$f (6) = - \sqrt{- 6 + 7} + 5$

Step 6.4.2.1.2

Add $- 6$ and $7$ .

$f (6) = - \sqrt{1} + 5$

Step 6.4.2.1.3

Any root of $1$ is $1$ .

$f (6) = - 1 \cdot 1 + 5$

Step 6.4.2.1.4

Multiply $- 1$ by $1$ .

$f (6) = - 1 + 5$

Step 6.4.2.2

Add $- 1$ and $5$ .

$f (6) = 4$

Step 6.4.2.3

The final answer is $4$ .

$y = 4$

Step 6.4.3

Convert $4$ to decimal.

$= 4$

Step 6.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 5 & 6.41 \\ 5 & 3.59 \\ 6 & 6 \\ 6 & 4 \\ 7 & 5 \end{array}$

Step 7

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(7, 5)$

Focus: $(\frac{27}{4}, 5)$

Axis of Symmetry: $y = 5$

Directrix: $x = \frac{29}{4}$

$\begin{array}{c} x & y \\ 5 & 6.41 \\ 5 & 3.59 \\ 6 & 6 \\ 6 & 4 \\ 7 & 5 \end{array}$

Step 8