The given equation is:

\[
3x - 7y = 15
\]

This is a **linear equation** in two variables, \(x\) and \(y\). To explore this equation further, we can perform different operations such as solving for one variable in terms of the other, finding its intercepts, or expressing it in slope-intercept form.

### Solving for \(y\) (Slope-Intercept Form):
To express the equation in slope-intercept form, which is \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept:

1. Start with:
\[
3x - 7y = 15
\]

2. Subtract \(3x\) from both sides:
\[
-7y = -3x + 15
\]

3. Divide the entire equation by \(-7\):
\[
y = \frac{3}{7}x - \frac{15}{7}
\]

Now, the equation is in the form \(y = \frac{3}{7}x - \frac{15}{7}\), where the slope \(m = \frac{3}{7}\) and the y-intercept is \(-\frac{15}{7}\).

Would you like further details on graphing or solving this in another way?

Here are some questions that expand on this topic:

1. How do you find the x-intercept of this equation?
2. What is the slope of a line perpendicular to this one?
3. How can you express the equation in standard form?
4. What is the significance of the slope in real-world scenarios?
5. How would the graph of this equation change if the constant term (15) changed?

**Tip:** When dealing with linear equations, always check if expressing it in different forms (slope-intercept, point-slope, etc.) helps better understand the problem.

Explore how to solve the linear equation 3x - 7y = 15 by expressing it in slope-intercept form. This solution demonstrates key algebraic concepts and methods, including how to find the slope and y-intercept. Suitable for students in Grades 8-10.

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