Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of .

Find the equation(s) of the tangent line(s) to the graph of the given equation at .
(Type an equation. Use a comma to separate answers as needed.)

Ask by Hills Tran. in the United States

Mar 24,2025

Question

Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of .

Find the equation(s) of the tangent line(s) to the graph of the given equation at .
(Type an equation. Use a comma to separate answers as needed.)

Ask by Hills Tran. in the United States

Mar 24,2025

UpStudy AI · Accepted Answer

To find the tangent line, we start with the given equation
$$
xy - x - 18 = 0.
$$

**Step 1. Find the point on the curve when $$ x = 3 $$.**

Substitute $$ x = 3 $$ into the equation:
$$
3y - 3 - 18 = 0 \quad \Longrightarrow \quad 3y - 21 = 0 \quad \Longrightarrow \quad 3y = 21 \quad \Longrightarrow \quad y = 7.
$$
So, the point is $$(3,7)$$.

**Step 2. Differentiate the equation implicitly with respect to $$ x $$.**

Differentiate each term:
$$
\frac{d}{dx}(xy) - \frac{d}{dx}(x) - \frac{d}{dx}(18) = 0.
$$
Using the product rule for $$ xy $$, we have:
$$
x\frac{dy}{dx} + y - 1 = 0.
$$
Solve for $$\frac{dy}{dx}$$:
$$
x\frac{dy}{dx} = 1 - y \quad \Longrightarrow \quad \frac{dy}{dx} = \frac{1 - y}{x}.
$$

**Step 3. Evaluate the derivative at the point $$(3,7)$$.**

Substitute $$ x = 3 $$ and $$ y = 7 $$:
$$
\frac{dy}{dx} = \frac{1 - 7}{3} = \frac{-6}{3} = -2.
$$
The slope of the tangent line at $$(3,7)$$ is $$-2$$.

**Step 4. Write the equation of the tangent line using the point-slope form.**

The point-slope form is:
$$
y - y_1 = m(x - x_1).
$$
Substitute $$ m = -2 $$ and $$(x_1, y_1) = (3, 7)$$:
$$
y - 7 = -2(x - 3).
$$

Simplify the equation:
$$
y - 7 = -2x + 6 \quad \Longrightarrow \quad y = -2x + 13.
$$

The equation of the tangent line is:
$$
y = -2x + 13.
$$
The equation of the tangent line at $$ x = 3 $$ is $$ y = -2x + 13 $$.

Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of .

Find the equation(s) of the tangent line(s) to the graph of the given equation at .
(Type an equation. Use a comma to separate answers as needed.)

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Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of . Find the equation(s) of the tangent line(s) to the graph of the given equation at . (Type an equation. Use a comma to separate answers as needed.)

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Find the equation(s) of the tangent line(s) to the graph of the indicated equation at the point(s) with the given value of .

Find the equation(s) of the tangent line(s) to the graph of the given equation at .
(Type an equation. Use a comma to separate answers as needed.)