To determine which graph represents the equation $y = \frac{2}{3}x + 1$, we can analyze the equation's slope and y-intercept and compare them to each graph:

1. Slope: The coefficient of $x$, which is $\frac{2}{3}$, represents the slope. A slope of $\frac{2}{3}$ means that for every 3 units increase in $x$, $y$ increases by 2 units. This is a positive slope, indicating an upward trend from left to right.

2. Y-Intercept: The constant term, $+1$, represents the y-intercept. This means the line should cross the y-axis at $y = 1$.

Now, let's examine each option:

• Option a: The line has a positive slope and intersects the y-axis at $y = 1$. This graph matches both the slope and y-intercept of the given equation.
• Option b: The line has a negative slope, which does not match the positive slope in the equation.
• Option c: The line also has a negative slope, so it does not match.
• Option d: The line appears to have a different slope and y-intercept from the equation.

The correct answer is therefore Option a, as it matches both the slope and y-intercept of the equation $y = \frac{2}{3}x + 1$.

Would you like more details on graphing linear equations or have any other questions?

Here are some related questions:

1. How can we determine the slope and y-intercept from different forms of equations?
2. What other methods can we use to graph linear equations?
3. How does changing the slope affect the appearance of a line?
4. What happens if we change the y-intercept in a linear equation?
5. How can we verify a point lies on a line given its equation?

Tip: Always check both the slope and y-intercept to ensure accuracy when matching an equation to its graph.