Find the equation of the tangent line to the curve $y = x^2 - 2x + 7$ which is parallel to the line $2x - y + 9 = 0$.

Step 1: Find the slope of the given line

To find the slope of the given line, we first rewrite its equation in the slope-intercept form, $y = mx + c$, where $m$ is the slope. Rearranging the terms of the equation $2x - y + 9 = 0$ gives us $y = 2x + 9$.

Step 2: Determine the slope of the tangent line

Since the tangent line is parallel to the given line, their slopes must be equal. From the previous step, we found the slope of the given line to be $2$. Therefore, the slope of the tangent line is also $2$.

Step 3: Find the derivative of the curve

The slope of the tangent to a curve at any point $(x, y)$ is given by its derivative, $\frac{dy}{dx}$. We differentiate the given equation of the curve, $y = x^2 - 2x + 7$, with respect to $x$.

Step 4: Find the point of tangency

We equate the derivative of the curve, which represents the slope of the tangent, to the slope of the tangent line we found earlier. This allows us to find the $x$ -coordinate of the point of tangency. Then, we substitute this $x$ -value back into the original curve equation to find the corresponding $y$ -coordinate.

Step 5: Write the equation of the tangent line

Now that we have the slope $m=2$ and the point of tangency $(x_1, y_1) = (2, 7)$, we can use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, to find the equation of the tangent line. We substitute the values and simplify to get the final equation.

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