What is an equation of the line that passes through the point (2,−3) and is parallel to the line 5x − 2y = 2

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!

What is an equation of the line that passes through the point (2,−3) and is parallel to the line 5x − 2y = 2?

Learn how to find the equation of a line that passes through the point (2,−3) and is parallel to 5x − 2y = 2. This step-by-step guide covers key algebraic concepts such as slope, parallel lines, and using point-slope form. Suitable for Grades 8-10.

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