To graph the line given by the equation:

$y + 3 = \frac{1}{2}(x - 7)$

let’s start by rewriting it in slope-intercept form ($y = mx + b$), which will make it easier to identify the slope and y-intercept.

Step-by-Step Solution

1. Distribute $\frac{1}{2}$ on the right side: $y + 3 = \frac{1}{2}x - \frac{7}{2}$

2. Isolate $y$ by subtracting 3 from both sides: $y = \frac{1}{2}x - \frac{7}{2} - 3$

3. Combine constants on the right: $y = \frac{1}{2}x - \frac{7}{2} - \frac{6}{2} = \frac{1}{2}x - \frac{13}{2}$

Now, the equation in slope-intercept form is: $y = \frac{1}{2}x - \frac{13}{2}$

Identifying Key Points

• Slope ($m$): $\frac{1}{2}$, which means the line rises by 1 unit for every 2 units it moves to the right.
• Y-intercept ($b$): $-\frac{13}{2}$ (or -6.5), so the line crosses the y-axis at $(0, -6.5)$.

Plotting the Line

1. Start at the y-intercept point: $(0, -6.5)$.

2. Use the slope to find another point:

   • From $(0, -6.5)$, go up 1 unit and right 2 units, reaching the point $(2, -5.5)$.

3. Draw a straight line through these points to complete the graph.

Would you like more details on any part of this solution?

Here are some questions to explore this further:

1. How can you convert any point-slope equation to slope-intercept form?
2. What does the slope $\frac{1}{2}$ tell us about the steepness and direction of the line?
3. How would you find the x-intercept of this line?
4. What other methods could be used to graph this line?
5. How would the graph change if the slope was negative?

Tip: When graphing, using multiple points helps ensure accuracy and a straight line.