Let's solve this by analyzing the graph.

Step 1: Find the slope of the line

The line passes through the points $(0, 4)$ (y-intercept) and $(2, -2)$. To find the slope $m$, use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the points $(0, 4)$ and $(2, -2)$: $m = \frac{-2 - 4}{2 - 0} = \frac{-6}{2} = -3$

Step 2: Write the equation of the line in slope-intercept form

The slope-intercept form of a line is: $y = mx + b$ Where $m = -3$ and $b = 4$ (from the y-intercept $(0, 4)$): $y = -3x + 4$

Step 3: Analyze the given options

Now that we know the slope is $-3$ and the y-intercept is 4, let's check the options:

1. $y = -3x + 4$: This is correct. The slope is $-3$ and the y-intercept is 4.
2. $y = 3x + 4$: Incorrect, the slope is positive here.
3. $y - 1 = -3(x - 1)$: Let's simplify this. Distribute the $-3$: $y - 1 = -3x + 3 \quad \Rightarrow \quad y = -3x + 4$ This matches the correct equation, so this is correct.
4. $y - 1 = 3(x - 1)$: Let's simplify this. Distribute the 3: $y - 1 = 3x - 3 \quad \Rightarrow \quad y = 3x - 2$ This does not match the correct equation, so this is incorrect.
5. $3x + y = 4$: Let's rearrange this to the slope-intercept form $y = mx + b$: $y = -3x + 4$ This matches the correct equation, so this is correct.

Correct answers:

• $y = -3x + 4$
• $y - 1 = -3(x - 1)$
• $3x + y = 4$

Would you like further details or clarifications? Here are some related questions:

1. How can we derive the equation of a line using two points?
2. What is the general form of a linear equation, and how does it relate to the slope-intercept form?
3. What happens to the slope when a line is vertical or horizontal?
4. How can you determine the slope from a graph without explicit points?
5. How do transformations (translations or scaling) affect the equation of a line?

Tip: Always confirm the slope of a line from its direction on the graph; a downward-sloping line has a negative slope.