Find an equation of the line parallel to y = −2/3x + 1
and that passes through the point (4, -5)

To find the equation of a line that is parallel to \( y = -\frac{2}{3}x + 1 \) and passes through the point \( (4, -5) \), we can follow these steps:

### 1. **Identify the slope**:
Parallel lines have the same slope. The given equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. 

In this case, the slope is:
\[
m = -\frac{2}{3}
\]

### 2. **Use the point-slope form of a line**:
The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( m \) is the slope of the line
- \( (x_1, y_1) \) is the point the line passes through, which in this case is \( (4, -5) \).

### 3. **Substitute the values**:
Now, substitute \( m = -\frac{2}{3} \), \( x_1 = 4 \), and \( y_1 = -5 \) into the point-slope form:
\[
y - (-5) = -\frac{2}{3}(x - 4)
\]
This simplifies to:
\[
y + 5 = -\frac{2}{3}(x - 4)
\]

### 4. **Simplify the equation**:
Distribute the slope on the right-hand side:
\[
y + 5 = -\frac{2}{3}x + \frac{8}{3}
\]

Now, subtract 5 from both sides to isolate \( y \):
\[
y = -\frac{2}{3}x + \frac{8}{3} - 5
\]

Convert 5 to a fraction with a denominator of 3:
\[
y = -\frac{2}{3}x + \frac{8}{3} - \frac{15}{3}
\]

Simplify the constants:
\[
y = -\frac{2}{3}x - \frac{7}{3}
\]

### Final equation of the line:
\[
y = -\frac{2}{3}x - \frac{7}{3}
\]

This is the equation of the line parallel to \( y = -\frac{2}{3}x + 1 \) and passing through the point \( (4, -5) \).

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Do you want more details or have any questions?

Here are 5 related questions to extend the topic:
1. What is the equation of a line perpendicular to \( y = -\frac{2}{3}x + 1 \) and passing through the point \( (4, -5) \)?
2. How would you find the x-intercept and y-intercept of this new line?
3. Can you convert the equation \( y = -\frac{2}{3}x - \frac{7}{3} \) into standard form?
4. What is the distance between the two parallel lines \( y = -\frac{2}{3}x + 1 \) and \( y = -\frac{2}{3}x - \frac{7}{3} \)?
5. How would you graph the equation \( y = -\frac{2}{3}x - \frac{7}{3} \)?

**Tip**: When finding the equation of a line, always check whether you need a parallel or perpendicular slope to ensure accuracy.

Find an equation of the line parallel to y = −2/3x + 1 and that passes through the point (4, -5)

Learn how to find the equation of a line that is parallel to y = -2/3x + 1 and passes through the point (4, -5). This solution explores key algebraic concepts such as parallel lines and the point-slope form of a line, suitable for students in Grades 8-10.

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