We are tasked with finding the equation of a line that is parallel to the line $y = 2x$ and passes through the point $(-2, 3)$.

Step 1: Understanding the slope of the parallel line

The equation $y = 2x$ is in slope-intercept form $y = mx + b$, where $m$ is the slope. In this case, the slope $m$ of the given line is 2.

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

Step 2: Using the point-slope form of a line

The point-slope form of a line is given by: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

We know:

• The slope $m = 2$,
• The point $(-2, 3)$.

Substituting these values into the point-slope form equation: $y - 3 = 2(x - (-2))$ Simplifying: $y - 3 = 2(x + 2)$ $y - 3 = 2x + 4$ $y = 2x + 7$

Step 3: Final equation

The equation of the line in slope-intercept form is: $y = 2x + 7$

This is the equation of the line that is parallel to $y = 2x$ and passes through the point $(-2, 3)$.

Would you like further explanation or a different approach?

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Here are five related questions to expand on this:

1. How do you convert a line's equation from slope-intercept form to general form?
2. What is the significance of parallel lines in geometry and algebra?
3. How would you find the equation of a line perpendicular to $y = 2x$?
4. How do you determine if two lines are parallel or perpendicular from their equations?
5. Can you graph the equation $y = 2x + 7$ and interpret its key features?

Tip: Remember that parallel lines always share the same slope, but their y-intercepts differ!