To find the equation of the line that passes through the point $(2, -3)$ and is parallel to the line given by the equation $5x - 2y = 2$, follow these steps:

Step 1: Find the slope of the given line

Start by rewriting the equation of the given line in slope-intercept form $y = mx + b$, where $m$ is the slope.

The equation is: $5x - 2y = 2$ Solve for $y$: $-2y = -5x + 2$ $y = \frac{5}{2}x - 1$ So, the slope $m$ of the given line is $\frac{5}{2}$.

Step 2: Use the slope of the parallel line

Since parallel lines have the same slope, the slope of the line we are looking for is also $\frac{5}{2}$.

Step 3: Use the point-slope form

Now, use the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is the point the line passes through, and $m$ is the slope. Here, the point is $(2, -3)$ and the slope is $\frac{5}{2}$.

Substitute the values into the formula: $y - (-3) = \frac{5}{2}(x - 2)$ $y + 3 = \frac{5}{2}(x - 2)$

Step 4: Simplify the equation

Distribute the slope: $y + 3 = \frac{5}{2}x - 5$ Now, subtract 3 from both sides: $y = \frac{5}{2}x - 5 - 3$ $y = \frac{5}{2}x - 8$

Final Answer:

The equation of the line is: $y = \frac{5}{2}x - 8$

Do you want further details or have any questions?

Here are some related questions:

1. How do you find the equation of a line given two points?
2. How do you find the slope of a perpendicular line?
3. What does the slope of a line represent geometrically?
4. How do you convert from slope-intercept form to standard form?
5. What is the significance of the y-intercept in the equation of a line?

Tip: When lines are parallel, their slopes are always equal!