The problem asks to form linear equations that pass through given pairs of points. Let's solve both parts step by step.

Part (a): Points (-1, 4) and (1, 0)

To find the equation of a line, we use the formula for the slope $m$ between two points:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For points $(-1, 4)$ and $(1, 0)$:

$m = \frac{0 - 4}{1 - (-1)} = \frac{-4}{2} = -2$

Now, using the point-slope form of the line equation, $y - y_1 = m(x - x_1)$, we can plug in one of the points, say $(-1, 4)$, and the slope $m = -2$:

$y - 4 = -2(x - (-1))$ $y - 4 = -2(x + 1)$ $y - 4 = -2x - 2$ $y = -2x + 2$

Thus, the equation for part (a) is: $y = -2x + 2$

Part (b): Points (1, 4) and (2, 3)

Again, using the slope formula for points $(1, 4)$ and $(2, 3)$:

$m = \frac{3 - 4}{2 - 1} = \frac{-1}{1} = -1$

Now, applying the point-slope form with point $(1, 4)$ and slope $m = -1$:

$y - 4 = -1(x - 1)$ $y - 4 = -1x + 1$ $y = -x + 5$

Thus, the equation for part (b) is: $y = -x + 5$

Do you need any further clarifications or more details on the solution?

Here are five questions that might help deepen your understanding:

1. How would you calculate the equation if the slope were zero?
2. What does the slope tell us about the direction of the line?
3. How can you find the x-intercept of the line from the equation?
4. How can you check if the given points lie on the derived line equation?
5. What if the points were the same? What kind of line would result?

Tip: Always verify your line equation by substituting both points into the final equation to ensure they satisfy it.