To graph the equation $y + 7 = -\frac{1}{2}(x + 7)$, we first need to simplify and rearrange it into the slope-intercept form, which is $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

Step-by-Step Simplification

1. Rewrite the equation:

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

2. Distribute the $-\frac{1}{2}$ on the right side:

   $y + 7 = -\frac{1}{2}x - \frac{1}{2} \times 7$

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

3. Subtract 7 from both sides to isolate $y$:

   $y = -\frac{1}{2}x - \frac{7}{2} - 7$

   Convert 7 to a fraction with a denominator of 2:

   $y = -\frac{1}{2}x - \frac{7}{2} - \frac{14}{2}$

4. Combine like terms:

   $y = -\frac{1}{2}x - \frac{21}{2}$

Graphing the Equation

The equation is now in slope-intercept form:

$y = -\frac{1}{2}x - \frac{21}{2}$

• Slope (m): $-\frac{1}{2}$
• Y-intercept (b): $-\frac{21}{2}$

How to Graph

1. Plot the y-intercept: Start by plotting the point $(0, -\frac{21}{2})$ or $(0, -10.5)$ on the y-axis.

2. Use the slope: The slope is $-\frac{1}{2}$, which means for every 1 unit increase in $x$, $y$ decreases by $\frac{1}{2}$. From the y-intercept:

   • Move right 2 units (positive direction on x-axis).
   • Move down 1 unit (negative direction on y-axis).

3. Draw the line: After plotting these points, draw a straight line through them, extending in both directions.

Visualization

Here is a rough description of how the graph looks:

• The line crosses the y-axis at $(0, -10.5)$.
• The line slopes downwards to the right, indicating a negative slope.

Would you like me to provide more details on any of these steps, or do you have any questions?

Related Questions

1. What are the x- and y-intercepts of the equation $y = -\frac{1}{2}x - \frac{21}{2}$?
2. How does the slope of the line affect its direction?
3. What are some methods to find the intersection points of two linear equations?
4. How can you determine if two lines are parallel or perpendicular?
5. How do you convert between different forms of a linear equation (e.g., slope-intercept, point-slope)?

Tip

When graphing lines, always start by plotting the y-intercept and then use the slope to find additional points. This method ensures accuracy and helps visualize the slope's direction (positive or negative).