To address the question in the uploaded image:

Part (a): Identifying Why $y = 2x - 3$ Does Not Match the Graph

1. Slope Mismatch: The rule $y = 2x - 3$ has a slope of 2, meaning it rises 2 units for every 1 unit it moves to the right. However, the graph shown has a negative slope (it decreases as $x$ increases), indicating the slope in the equation does not match the graph's slope.
2. Y-Intercept Mismatch: The rule $y = 2x - 3$ has a y-intercept of $-3$ (where $y$-value is $-3$ when $x = 0$). The graph's y-intercept is at $y = 4$, so the intercept in the equation does not match the graph.

Part (b): Alternative Rule Matching the Graph

To find an alternative rule:

1. Identify the Slope: From the graph, the slope can be calculated. For every decrease of 4 units in $y$, $x$ increases by 4 units, so the slope is: $\text{slope} = \frac{-4}{4} = -1$
2. Identify the Y-Intercept: The graph intersects the y-axis at $y = 4$. Hence, the y-intercept is 4.

The rule matching the graph is: $y = -x + 4$

Would you like more details or have additional questions?

Related Questions:

1. How do you calculate the slope of a line given two points on the graph?
2. Can a line's equation be written in forms other than slope-intercept form? If so, how?
3. What is the effect of changing the slope in a linear equation on the graph?
4. How does altering the y-intercept in a linear equation affect its position on the graph?
5. What techniques are useful for determining if a point lies on a given line?

Tip:

When examining graphs and equations, always match the slope and y-intercept to confirm correspondence.