Graph x-y=9 7x^2+3y^2-24y=139

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

Step 1

Graph $x - y = 9$ .

Solve for $y$ .

Subtract $x$ from both sides of the equation.

$- y = 9 - x$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $9$ by $- 1$ .

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

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

$y = - 9 + x$

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Reorder $- 9$ and $x$ .

$y = x - 9$

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

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

$m = 1$

$b = - 9$

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: $(0, - 9)$

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

Reorder $- 9$ and $x$ .

$y = x - 9$

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

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

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

Find the standard form of the ellipse.

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

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

$a = 3$

$b = - 24$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

Simplify the right side.

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

Factor $2$ out of $- 24$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $3$ out of $- 12$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 4$ by $1$ .

$d = - 4$

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

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $3$ .

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

Divide $576$ by $12$ .

$e = 0 - 1 \cdot 48$

Multiply $- 1$ by $48$ .

$e = 0 - 48$

Subtract $48$ from $0$ .

$e = - 48$

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

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

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

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

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

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

Add $139$ and $48$ .

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

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

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

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 $1$ .

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

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.

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

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.

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

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

$k = 4$

$h = 0$

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

$(0, 4)$

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

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

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

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

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

Simplify.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Factor $3$ out of $561$ .

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Factor $7$ out of $1309$ .

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

Cancel the common factors.

Factor $7$ out of $49$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

Multiply $3$ by $7$ .

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

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

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

Multiply $7$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $187$ by $7$ .

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

Multiply $- 187$ by $3$ .

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

Subtract $561$ from $1309$ .

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

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

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

Simplify the numerator.

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

Factor $4$ out of $748$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $21$ by $187$ .

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

Find the vertices.

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

$(h, k + a)$

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

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

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

$(h, k - a)$

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

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

Simplify.

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

Ellipses have two vertices.

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

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

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

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

Find the foci.

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

$(h, k + c)$

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

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

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

$(h, k - c)$

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

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

Simplify.

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

Ellipses have two foci.

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

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

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

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

Find the eccentricity.

Find the eccentricity by using the following formula.

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

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

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

Simplify.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Factor $3$ out of $561$ .

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

Factor $7$ out of $1309$ .

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

Cancel the common factors.

Factor $7$ out of $49$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

Multiply $3$ by $7$ .

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

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

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

Multiply $7$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $187$ by $7$ .

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

Multiply $- 187$ by $3$ .

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

Subtract $561$ from $1309$ .

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

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

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

Simplify the numerator.

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

Factor $4$ out of $748$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $3$ .

Factor $3$ out of $21$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine using the product rule for radicals.

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

Multiply $21$ by $187$ .

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

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

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

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

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

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

Factor $561$ out of $3927$ .

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

Cancel the common factors.

Factor $561$ out of $561$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $7$ by $1$ .

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

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

Graph x-y=9 7x^2+3y^2-24y=139

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