To find the equation of the line that is parallel to $y = 2x - 4$ and passes through the point $(2,5)$, we follow these steps:

Step 1: Identify the slope of the given line

The equation of the given line is: $y = 2x - 4$ This is in slope-intercept form, $y = mx + b$, where $m$ is the slope of the line. Here, the slope $m$ is $2$.

Since parallel lines have the same slope, the line we are looking for will also have a slope of $2$.

Step 2: Use the point-slope form

To find the equation of a line when we know its slope and a point it passes through, we can use the point-slope form of a line: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

Here:

• The slope $m = 2$,
• The point $(x_1, y_1) = (2,5)$.

Substitute these values into the point-slope form: $y - 5 = 2(x - 2)$

Step 3: Simplify the equation

Now, distribute the $2$ on the right-hand side: $y - 5 = 2x - 4$ Next, add $5$ to both sides to solve for $y$: $y = 2x + 1$

Final Answer

The equation of the line that is parallel to $y = 2x - 4$ and passes through the point $(2,5)$ is: $y = 2x + 1$

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Related Questions:

1. How would you find the equation of a line perpendicular to $y = 2x - 4$ that passes through $(2,5)$?
2. What is the slope of a line parallel to $y = -3x + 7$?
3. If a line passes through $(3, -2)$ and has a slope of $-1$, what is its equation?
4. How can you verify if two lines are parallel?
5. What is the slope-intercept form of a line passing through $(0,4)$ and $(2,8)$?

Tip: In point-slope form, use any point on the line along with the slope to derive the line equation.