Algebra Examples

$x - y = 9$ $7 x^{2} + 3 y^{2} - 24 y = 139$

Step 1

Graph $x - y = 9$ .

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Step 1.1

Solve for $y$ .

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$- y = 9 - x$

Step 1.1.2

Divide each term in $- y = 9 - x$ by $- 1$ and simplify.

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Step 1.1.2.1

Divide each term in $- y = 9 - x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{9}{- 1} + \frac{- x}{- 1}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{9}{- 1} + \frac{- x}{- 1}$

Step 1.1.2.2.2

Divide $y$ by $1$ .

$y = \frac{9}{- 1} + \frac{- x}{- 1}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Divide $9$ by $- 1$ .

$y = - 9 + \frac{- x}{- 1}$

Step 1.1.2.3.1.2

Dividing two negative values results in a positive value.

$y = - 9 + \frac{x}{1}$

Step 1.1.2.3.1.3

Divide $x$ by $1$ .

$y = - 9 + x$

Step 1.2

Rewrite in slope-intercept form.

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Step 1.2.1

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 1.2.2

Reorder $- 9$ and $x$ .

$y = x - 9$

Step 1.3

Use the slope-intercept form to find the slope and y-intercept.

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Step 1.3.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = 1$

$b = - 9$

Step 1.3.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $1$

y-intercept: $(0, - 9)$

Slope: $1$

y-intercept: $(0, - 9)$

Step 1.4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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Step 1.4.1

Reorder $- 9$ and $x$ .

$y = x - 9$

Step 1.4.2

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ 0 & - 9 \\ 1 & - 8 \end{array}$

Step 1.5

Graph the line using the slope and the y-intercept, or the points.

Slope: $1$

y-intercept: $(0, - 9)$

$\begin{array}{c} x & y \\ 0 & - 9 \\ 1 & - 8 \end{array}$

Slope: $1$

y-intercept: $(0, - 9)$

$\begin{array}{c} x & y \\ 0 & - 9 \\ 1 & - 8 \end{array}$

Step 2

Find the properties of the conic section $7 x^{2} + 3 y^{2} - 24 y = 139$ .

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Step 2.1

Find the standard form of the ellipse.

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Step 2.1.1

Complete the square for $3 y^{2} - 24 y$ .

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Step 2.1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 3$

$b = - 24$

$c = 0$

Step 2.1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 24}{2 \cdot 3}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Cancel the common factor of $- 24$ and $2$ .

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Step 2.1.1.3.2.1.1

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot 3}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

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Step 2.1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot - 12}{2 (3)}$

Step 2.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot 3}$

Step 2.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 12}{3}$

Step 2.1.1.3.2.2

Cancel the common factor of $- 12$ and $3$ .

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Step 2.1.1.3.2.2.1

Factor $3$ out of $- 12$ .

$d = \frac{3 \cdot - 4}{3}$

Step 2.1.1.3.2.2.2

Cancel the common factors.

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Step 2.1.1.3.2.2.2.1

Factor $3$ out of $3$ .

$d = \frac{3 \cdot - 4}{3 (1)}$

Step 2.1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot - 4}{3 \cdot 1}$

Step 2.1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 4}{1}$

Step 2.1.1.3.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

Step 2.1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot 3}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Raise $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot 3}$

Step 2.1.1.4.2.1.2

Multiply $4$ by $3$ .

$e = 0 - \frac{576}{12}$

Step 2.1.1.4.2.1.3

Divide $576$ by $12$ .

$e = 0 - 1 \cdot 48$

Step 2.1.1.4.2.1.4

Multiply $- 1$ by $48$ .

$e = 0 - 48$

Step 2.1.1.4.2.2

Subtract $48$ from $0$ .

$e = - 48$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y - 4)}^{2} - 48$ .

$3 {(y - 4)}^{2} - 48$

Step 2.1.2

Substitute $3 {(y - 4)}^{2} - 48$ for $3 y^{2} - 24 y$ in the equation $7 x^{2} + 3 y^{2} - 24 y = 139$ .

$7 x^{2} + 3 {(y - 4)}^{2} - 48 = 139$

Step 2.1.3

Move $- 48$ to the right side of the equation by adding $48$ to both sides.

$7 x^{2} + 3 {(y - 4)}^{2} = 139 + 48$

Step 2.1.4

Add $139$ and $48$ .

$7 x^{2} + 3 {(y - 4)}^{2} = 187$

Step 2.1.5

Divide each term by $187$ to make the right side equal to one.

$\frac{7 x^{2}}{187} + \frac{3 {(y - 4)}^{2}}{187} = \frac{187}{187}$

Step 2.1.6

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{x^{2}}{\frac{187}{7}} + \frac{{(y - 4)}^{2}}{\frac{187}{3}} = 1$

Step 2.2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Step 2.3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = \frac{\sqrt{561}}{3}$

$b = \frac{\sqrt{1309}}{7}$

$k = 4$

$h = 0$

Step 2.4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 4)$

Step 2.5

Find $c$ , the distance from the center to a focus.

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Step 2.5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Step 2.5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(\frac{\sqrt{561}}{3})}^{2} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Apply the product rule to $\frac{\sqrt{561}}{3}$ .

$\sqrt{\frac{{\sqrt{561}}^{2}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.2

Rewrite ${\sqrt{561}}^{2}$ as $561$ .

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Step 2.5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{561}$ as $561^{\frac{1}{2}}$ .

$\sqrt{\frac{{(561^{\frac{1}{2}})}^{2}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{561^{\frac{1}{2} \cdot 2}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{561^{\frac{2}{2}}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.2.4

Cancel the common factor of $2$ .

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Step 2.5.3.2.4.1

Cancel the common factor.

$\sqrt{\frac{561^{\frac{2}{2}}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.2.4.2

Rewrite the expression.

$\sqrt{\frac{561^{1}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.2.5

Evaluate the exponent.

$\sqrt{\frac{561}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.3

Raise $3$ to the power of $2$ .

$\sqrt{\frac{561}{9} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.4

Cancel the common factor of $561$ and $9$ .

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Step 2.5.3.4.1

Factor $3$ out of $561$ .

$\sqrt{\frac{3 (187)}{9} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.4.2

Cancel the common factors.

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Step 2.5.3.4.2.1

Factor $3$ out of $9$ .

$\sqrt{\frac{3 \cdot 187}{3 \cdot 3} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.4.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 187}{3 \cdot 3} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.4.2.3

Rewrite the expression.

$\sqrt{\frac{187}{3} - {(\frac{\sqrt{1309}}{7})}^{2}}$

Step 2.5.3.5

Apply the product rule to $\frac{\sqrt{1309}}{7}$ .

$\sqrt{\frac{187}{3} - \frac{{\sqrt{1309}}^{2}}{7^{2}}}$

Step 2.5.3.6

Rewrite ${\sqrt{1309}}^{2}$ as $1309$ .

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Step 2.5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1309}$ as $1309^{\frac{1}{2}}$ .

$\sqrt{\frac{187}{3} - \frac{{(1309^{\frac{1}{2}})}^{2}}{7^{2}}}$

Step 2.5.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{187}{3} - \frac{1309^{\frac{1}{2} \cdot 2}}{7^{2}}}$

Step 2.5.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{187}{3} - \frac{1309^{\frac{2}{2}}}{7^{2}}}$

Step 2.5.3.6.4

Cancel the common factor of $2$ .

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Step 2.5.3.6.4.1

Cancel the common factor.

$\sqrt{\frac{187}{3} - \frac{1309^{\frac{2}{2}}}{7^{2}}}$

Step 2.5.3.6.4.2

Rewrite the expression.

$\sqrt{\frac{187}{3} - \frac{1309^{1}}{7^{2}}}$

Step 2.5.3.6.5

Evaluate the exponent.

$\sqrt{\frac{187}{3} - \frac{1309}{7^{2}}}$

Step 2.5.3.7

Raise $7$ to the power of $2$ .

$\sqrt{\frac{187}{3} - \frac{1309}{49}}$

Step 2.5.3.8

Cancel the common factor of $1309$ and $49$ .

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Step 2.5.3.8.1

Factor $7$ out of $1309$ .

$\sqrt{\frac{187}{3} - \frac{7 (187)}{49}}$

Step 2.5.3.8.2

Cancel the common factors.

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Step 2.5.3.8.2.1

Factor $7$ out of $49$ .

$\sqrt{\frac{187}{3} - \frac{7 \cdot 187}{7 \cdot 7}}$

Step 2.5.3.8.2.2

Cancel the common factor.

$\sqrt{\frac{187}{3} - \frac{7 \cdot 187}{7 \cdot 7}}$

Step 2.5.3.8.2.3

Rewrite the expression.

$\sqrt{\frac{187}{3} - \frac{187}{7}}$

Step 2.5.3.9

To write $\frac{187}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\sqrt{\frac{187}{3} \cdot \frac{7}{7} - \frac{187}{7}}$

Step 2.5.3.10

To write $- \frac{187}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{187}{3} \cdot \frac{7}{7} - \frac{187}{7} \cdot \frac{3}{3}}$

Step 2.5.3.11

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.3.11.1

Multiply $\frac{187}{3}$ by $\frac{7}{7}$ .

$\sqrt{\frac{187 \cdot 7}{3 \cdot 7} - \frac{187}{7} \cdot \frac{3}{3}}$

Step 2.5.3.11.2

Multiply $3$ by $7$ .

$\sqrt{\frac{187 \cdot 7}{21} - \frac{187}{7} \cdot \frac{3}{3}}$

Step 2.5.3.11.3

Multiply $\frac{187}{7}$ by $\frac{3}{3}$ .

$\sqrt{\frac{187 \cdot 7}{21} - \frac{187 \cdot 3}{7 \cdot 3}}$

Step 2.5.3.11.4

Multiply $7$ by $3$ .

$\sqrt{\frac{187 \cdot 7}{21} - \frac{187 \cdot 3}{21}}$

Step 2.5.3.12

Combine the numerators over the common denominator.

$\sqrt{\frac{187 \cdot 7 - 187 \cdot 3}{21}}$

Step 2.5.3.13

Simplify the numerator.

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Step 2.5.3.13.1

Multiply $187$ by $7$ .

$\sqrt{\frac{1309 - 187 \cdot 3}{21}}$

Step 2.5.3.13.2

Multiply $- 187$ by $3$ .

$\sqrt{\frac{1309 - 561}{21}}$

Step 2.5.3.13.3

Subtract $561$ from $1309$ .

$\sqrt{\frac{748}{21}}$

Step 2.5.3.14

Rewrite $\sqrt{\frac{748}{21}}$ as $\frac{\sqrt{748}}{\sqrt{21}}$ .

$\frac{\sqrt{748}}{\sqrt{21}}$

Step 2.5.3.15

Simplify the numerator.

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Step 2.5.3.15.1

Rewrite $748$ as $2^{2} \cdot 187$ .

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Step 2.5.3.15.1.1

Factor $4$ out of $748$ .

$\frac{\sqrt{4 (187)}}{\sqrt{21}}$

Step 2.5.3.15.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 187}}{\sqrt{21}}$

Step 2.5.3.15.2

Pull terms out from under the radical.

$\frac{2 \sqrt{187}}{\sqrt{21}}$

Step 2.5.3.16

Multiply $\frac{2 \sqrt{187}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{2 \sqrt{187}}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}$

Step 2.5.3.17

Combine and simplify the denominator.

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Step 2.5.3.17.1

Multiply $\frac{2 \sqrt{187}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{2 \sqrt{187} \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 2.5.3.17.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{1} \sqrt{21}}$

Step 2.5.3.17.3

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{1} {\sqrt{21}}^{1}}$

Step 2.5.3.17.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 2.5.3.17.5

Add $1$ and $1$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{2}}$

Step 2.5.3.17.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.5.3.17.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 2.5.3.17.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{187} \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 2.5.3.17.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{187} \sqrt{21}}{21^{\frac{2}{2}}}$

Step 2.5.3.17.6.4

Cancel the common factor of $2$ .

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Step 2.5.3.17.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{187} \sqrt{21}}{21^{\frac{2}{2}}}$

Step 2.5.3.17.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{187} \sqrt{21}}{21^{1}}$

Step 2.5.3.17.6.5

Evaluate the exponent.

$\frac{2 \sqrt{187} \sqrt{21}}{21}$

Step 2.5.3.18

Simplify the numerator.

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Step 2.5.3.18.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{21 \cdot 187}}{21}$

Step 2.5.3.18.2

Multiply $21$ by $187$ .

$\frac{2 \sqrt{3927}}{21}$

Step 2.6

Find the vertices.

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Step 2.6.1

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 2.6.2

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

$(0, 4 + \frac{\sqrt{561}}{3})$

Step 2.6.3

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 2.6.4

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

$(0, 4 - (\frac{\sqrt{561}}{3}))$

Step 2.6.5

Simplify.

$(0, 4 - \frac{\sqrt{561}}{3})$

Step 2.6.6

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 4 + \frac{\sqrt{561}}{3})$

${Vertex}_{2}$ : $(0, 4 - \frac{\sqrt{561}}{3})$

${Vertex}_{1}$ : $(0, 4 + \frac{\sqrt{561}}{3})$

${Vertex}_{2}$ : $(0, 4 - \frac{\sqrt{561}}{3})$

Step 2.7

Find the foci.

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Step 2.7.1

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 2.7.2

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

$(0, 4 + \frac{2 \sqrt{3927}}{21})$

Step 2.7.3

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 2.7.4

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

$(0, 4 - (\frac{2 \sqrt{3927}}{21}))$

Step 2.7.5

Simplify.

$(0, 4 - \frac{2 \sqrt{3927}}{21})$

Step 2.7.6

Ellipses have two foci.

${Focus}_{1}$ : $(0, 4 + \frac{2 \sqrt{3927}}{21})$

${Focus}_{2}$ : $(0, 4 - \frac{2 \sqrt{3927}}{21})$

${Focus}_{1}$ : $(0, 4 + \frac{2 \sqrt{3927}}{21})$

${Focus}_{2}$ : $(0, 4 - \frac{2 \sqrt{3927}}{21})$

Step 2.8

Find the eccentricity.

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Step 2.8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Step 2.8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(\frac{\sqrt{561}}{3})}^{2} - {(\frac{\sqrt{1309}}{7})}^{2}}}{\frac{\sqrt{561}}{3}}$

Step 2.8.3

Simplify.

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Step 2.8.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{\sqrt{561}}{3})}^{2} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.2

Apply the product rule to $\frac{\sqrt{561}}{3}$ .

$\sqrt{\frac{{\sqrt{561}}^{2}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.3

Rewrite ${\sqrt{561}}^{2}$ as $561$ .

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Step 2.8.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{561}$ as $561^{\frac{1}{2}}$ .

$\sqrt{\frac{{(561^{\frac{1}{2}})}^{2}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{561^{\frac{1}{2} \cdot 2}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.3.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{561^{\frac{2}{2}}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.3.4

Cancel the common factor of $2$ .

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Step 2.8.3.3.4.1

Cancel the common factor.

$\sqrt{\frac{561^{\frac{2}{2}}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.3.4.2

Rewrite the expression.

$\sqrt{\frac{561^{1}}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.3.5

Evaluate the exponent.

$\sqrt{\frac{561}{3^{2}} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.4

Raise $3$ to the power of $2$ .

$\sqrt{\frac{561}{9} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.5

Cancel the common factor of $561$ and $9$ .

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Step 2.8.3.5.1

Factor $3$ out of $561$ .

$\sqrt{\frac{3 (187)}{9} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.5.2

Cancel the common factors.

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Step 2.8.3.5.2.1

Factor $3$ out of $9$ .

$\sqrt{\frac{3 \cdot 187}{3 \cdot 3} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.5.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 187}{3 \cdot 3} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.5.2.3

Rewrite the expression.

$\sqrt{\frac{187}{3} - {(\frac{\sqrt{1309}}{7})}^{2}} \frac{3}{\sqrt{561}}$

Step 2.8.3.6

Apply the product rule to $\frac{\sqrt{1309}}{7}$ .

$\sqrt{\frac{187}{3} - \frac{{\sqrt{1309}}^{2}}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.7

Rewrite ${\sqrt{1309}}^{2}$ as $1309$ .

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Step 2.8.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1309}$ as $1309^{\frac{1}{2}}$ .

$\sqrt{\frac{187}{3} - \frac{{(1309^{\frac{1}{2}})}^{2}}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{187}{3} - \frac{1309^{\frac{1}{2} \cdot 2}}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.7.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{187}{3} - \frac{1309^{\frac{2}{2}}}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.7.4

Cancel the common factor of $2$ .

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Step 2.8.3.7.4.1

Cancel the common factor.

$\sqrt{\frac{187}{3} - \frac{1309^{\frac{2}{2}}}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.7.4.2

Rewrite the expression.

$\sqrt{\frac{187}{3} - \frac{1309^{1}}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.7.5

Evaluate the exponent.

$\sqrt{\frac{187}{3} - \frac{1309}{7^{2}}} \frac{3}{\sqrt{561}}$

Step 2.8.3.8

Raise $7$ to the power of $2$ .

$\sqrt{\frac{187}{3} - \frac{1309}{49}} \frac{3}{\sqrt{561}}$

Step 2.8.3.9

Cancel the common factor of $1309$ and $49$ .

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Step 2.8.3.9.1

Factor $7$ out of $1309$ .

$\sqrt{\frac{187}{3} - \frac{7 (187)}{49}} \frac{3}{\sqrt{561}}$

Step 2.8.3.9.2

Cancel the common factors.

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Step 2.8.3.9.2.1

Factor $7$ out of $49$ .

$\sqrt{\frac{187}{3} - \frac{7 \cdot 187}{7 \cdot 7}} \frac{3}{\sqrt{561}}$

Step 2.8.3.9.2.2

Cancel the common factor.

$\sqrt{\frac{187}{3} - \frac{7 \cdot 187}{7 \cdot 7}} \frac{3}{\sqrt{561}}$

Step 2.8.3.9.2.3

Rewrite the expression.

$\sqrt{\frac{187}{3} - \frac{187}{7}} \frac{3}{\sqrt{561}}$

Step 2.8.3.10

To write $\frac{187}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\sqrt{\frac{187}{3} \cdot \frac{7}{7} - \frac{187}{7}} \frac{3}{\sqrt{561}}$

Step 2.8.3.11

To write $- \frac{187}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{187}{3} \cdot \frac{7}{7} - \frac{187}{7} \cdot \frac{3}{3}} \frac{3}{\sqrt{561}}$

Step 2.8.3.12

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.8.3.12.1

Multiply $\frac{187}{3}$ by $\frac{7}{7}$ .

$\sqrt{\frac{187 \cdot 7}{3 \cdot 7} - \frac{187}{7} \cdot \frac{3}{3}} \frac{3}{\sqrt{561}}$

Step 2.8.3.12.2

Multiply $3$ by $7$ .

$\sqrt{\frac{187 \cdot 7}{21} - \frac{187}{7} \cdot \frac{3}{3}} \frac{3}{\sqrt{561}}$

Step 2.8.3.12.3

Multiply $\frac{187}{7}$ by $\frac{3}{3}$ .

$\sqrt{\frac{187 \cdot 7}{21} - \frac{187 \cdot 3}{7 \cdot 3}} \frac{3}{\sqrt{561}}$

Step 2.8.3.12.4

Multiply $7$ by $3$ .

$\sqrt{\frac{187 \cdot 7}{21} - \frac{187 \cdot 3}{21}} \frac{3}{\sqrt{561}}$

Step 2.8.3.13

Combine the numerators over the common denominator.

$\sqrt{\frac{187 \cdot 7 - 187 \cdot 3}{21}} \frac{3}{\sqrt{561}}$

Step 2.8.3.14

Simplify the numerator.

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Step 2.8.3.14.1

Multiply $187$ by $7$ .

$\sqrt{\frac{1309 - 187 \cdot 3}{21}} \frac{3}{\sqrt{561}}$

Step 2.8.3.14.2

Multiply $- 187$ by $3$ .

$\sqrt{\frac{1309 - 561}{21}} \frac{3}{\sqrt{561}}$

Step 2.8.3.14.3

Subtract $561$ from $1309$ .

$\sqrt{\frac{748}{21}} \frac{3}{\sqrt{561}}$

Step 2.8.3.15

Rewrite $\sqrt{\frac{748}{21}}$ as $\frac{\sqrt{748}}{\sqrt{21}}$ .

$\frac{\sqrt{748}}{\sqrt{21}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.16

Simplify the numerator.

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Step 2.8.3.16.1

Rewrite $748$ as $2^{2} \cdot 187$ .

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Step 2.8.3.16.1.1

Factor $4$ out of $748$ .

$\frac{\sqrt{4 (187)}}{\sqrt{21}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.16.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 187}}{\sqrt{21}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.16.2

Pull terms out from under the radical.

$\frac{2 \sqrt{187}}{\sqrt{21}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.17

Multiply $\frac{2 \sqrt{187}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{2 \sqrt{187}}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}} \frac{3}{\sqrt{561}}$

Step 2.8.3.18

Combine and simplify the denominator.

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Step 2.8.3.18.1

Multiply $\frac{2 \sqrt{187}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{2 \sqrt{187} \sqrt{21}}{\sqrt{21} \sqrt{21}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{1} \sqrt{21}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.3

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{1} {\sqrt{21}}^{1}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{1 + 1}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.5

Add $1$ and $1$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{\sqrt{21}}^{2}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.8.3.18.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{2 \sqrt{187} \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{187} \sqrt{21}}{21^{\frac{1}{2} \cdot 2}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{187} \sqrt{21}}{21^{\frac{2}{2}}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.6.4

Cancel the common factor of $2$ .

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Step 2.8.3.18.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{187} \sqrt{21}}{21^{\frac{2}{2}}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{187} \sqrt{21}}{21^{1}} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.18.6.5

Evaluate the exponent.

$\frac{2 \sqrt{187} \sqrt{21}}{21} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.19

Cancel the common factor of $3$ .

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Step 2.8.3.19.1

Factor $3$ out of $21$ .

$\frac{2 \sqrt{187} \sqrt{21}}{3 (7)} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.19.2

Cancel the common factor.

$\frac{2 \sqrt{187} \sqrt{21}}{3 \cdot 7} \cdot \frac{3}{\sqrt{561}}$

Step 2.8.3.19.3

Rewrite the expression.

$\frac{2 \sqrt{187} \sqrt{21}}{7} \cdot \frac{1}{\sqrt{561}}$

Step 2.8.3.20

Combine using the product rule for radicals.

$\frac{2 \sqrt{21 \cdot 187}}{7} \cdot \frac{1}{\sqrt{561}}$

Step 2.8.3.21

Multiply $21$ by $187$ .

$\frac{2 \sqrt{3927}}{7} \cdot \frac{1}{\sqrt{561}}$

Step 2.8.3.22

Multiply $\frac{2 \sqrt{3927}}{7}$ by $\frac{1}{\sqrt{561}}$ .

$\frac{2 \sqrt{3927}}{7 \sqrt{561}}$

Step 2.8.3.23

Combine $\sqrt{3927}$ and $\sqrt{561}$ into a single radical.

$\frac{2 \sqrt{\frac{3927}{561}}}{7}$

Step 2.8.3.24

Cancel the common factor of $3927$ and $561$ .

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Step 2.8.3.24.1

Factor $561$ out of $3927$ .

$\frac{2 \sqrt{\frac{561 \cdot 7}{561}}}{7}$

Step 2.8.3.24.2

Cancel the common factors.

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Step 2.8.3.24.2.1

Factor $561$ out of $561$ .

$\frac{2 \sqrt{\frac{561 \cdot 7}{561 (1)}}}{7}$

Step 2.8.3.24.2.2

Cancel the common factor.

$\frac{2 \sqrt{\frac{561 \cdot 7}{561 \cdot 1}}}{7}$

Step 2.8.3.24.2.3

Rewrite the expression.

$\frac{2 \sqrt{\frac{7}{1}}}{7}$

Step 2.8.3.24.2.4

Divide $7$ by $1$ .

$\frac{2 \sqrt{7}}{7}$

Step 2.9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(0, 4)$

${Vertex}_{1}$ : $(0, 4 + \frac{\sqrt{561}}{3})$

${Vertex}_{2}$ : $(0, 4 - \frac{\sqrt{561}}{3})$

${Focus}_{1}$ : $(0, 4 + \frac{2 \sqrt{3927}}{21})$

${Focus}_{2}$ : $(0, 4 - \frac{2 \sqrt{3927}}{21})$

Eccentricity: $\frac{2 \sqrt{7}}{7}$

Center: $(0, 4)$

${Vertex}_{1}$ : $(0, 4 + \frac{\sqrt{561}}{3})$

${Vertex}_{2}$ : $(0, 4 - \frac{\sqrt{561}}{3})$

${Focus}_{1}$ : $(0, 4 + \frac{2 \sqrt{3927}}{21})$

${Focus}_{2}$ : $(0, 4 - \frac{2 \sqrt{3927}}{21})$

Eccentricity: $\frac{2 \sqrt{7}}{7}$

Step 3

Plot each graph on the same coordinate system.

$x - y = 9$

$7 x^{2} + 3 y^{2} - 24 y = 139$

Step 4