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The problem is to find the slope-intercept form of the equation of a line that is parallel to the given line \(3x + 7y = 14\) and passes through the point \((0, -5)\).

### Step 1: Find the slope of the given line.
To find the slope, we need to rewrite the equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Starting with:
\[
3x + 7y = 14
\]

Rearrange the equation to solve for \(y\):
\[
7y = -3x + 14
\]
\[
y = -\frac{3}{7}x + 2
\]

The slope of this line is \(-\frac{3}{7}\).

### Step 2: Use the slope to find the equation of the parallel line.
Since parallel lines have the same slope, the new line will also have a slope of \(-\frac{3}{7}\). The equation of a line in slope-intercept form is given by:
\[
y = mx + b
\]

We need to find \(b\), the y-intercept, using the point \((0, -5)\):
\[
-5 = -\frac{3}{7}(0) + b
\]
\[
b = -5
\]

### Step 3: Write the equation of the line.
The equation of the line parallel to \(3x + 7y = 14\) and passing through the point \((0, -5)\) is:
\[
y = -\frac{3}{7}x - 5
\]

Would you like further details or have any questions?

Here are five related questions for further practice:
1. How would you find the equation of a line perpendicular to \(3x + 7y = 14\) and passing through \((0, -5)\)?
2. What is the slope of a line that is perpendicular to \(y = -\frac{3}{7}x + 2\)?
3. How can you determine if two lines are parallel, perpendicular, or neither?
4. If a line passes through the points \((1, 2)\) and \((4, -1)\), what is its slope?
5. How would you convert the standard form \(Ax + By = C\) to slope-intercept form?

**Tip:** Remember that parallel lines have identical slopes, while perpendicular lines have slopes that are negative reciprocals of each other.

Consider the line 3x + 7y = 14. Find the slope-intercept form of the equation of a line that is parallel to the given line and passes through the point (0, -5).

Solve the linear algebra problem of finding the slope-intercept form of a line that is parallel to the given line 3x + 7y = 14 and passes through the point (0, -5). Learn key concepts such as the slope-intercept form, calculating slopes, and applying the slope for parallel lines. This guide is perfect for students in grades 8-10.

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