To determine the intercepts of the line given by the equation:

$-4x + 7y = 3$

we need to find both the $x$-intercept and the $y$-intercept.

Finding the $x$-intercept

The $x$-intercept occurs where $y = 0$. Substituting $y = 0$ into the equation:

$-4x + 7(0) = 3$

$-4x = 3$

$x = -\frac{3}{4}$

So, the $x$-intercept is:

$\left(-\frac{3}{4}, 0\right)$

Finding the $y$-intercept

The $y$-intercept occurs where $x = 0$. Substituting $x = 0$ into the equation:

$-4(0) + 7y = 3$

$7y = 3$

$y = \frac{3}{7}$

So, the $y$-intercept is:

$\left(0, \frac{3}{7}\right)$

Final Answer

• $x$-intercept: $\left(-\frac{3}{4}, 0\right)$
• $y$-intercept: $\left(0, \frac{3}{7}\right)$

Would you like further explanation or have any questions?

---

Related Questions:

1. How do you interpret the $x$-intercept and $y$-intercept geometrically on the coordinate plane?
2. What is the importance of intercepts in solving linear equations graphically?
3. How do you determine the intercepts if the line equation is in slope-intercept form $y = mx + b$?
4. Can a line have both intercepts as zero? What would that mean?
5. How would you find the intercepts for a line parallel to the given line?

Tip: Remember, the $x$-intercept is found by setting $y = 0$ and solving for $x$, while the $y$-intercept is found by setting $x = 0$ and solving for $y$.