Certainly! To determine the correct graph for the linear equation [tex]\( y + 3 = -\frac{2}{3}(x - 4) \)[/tex], we should follow these steps to rewrite and understand the equation.
### Step 1: Rewrite in Slope-Intercept Form
To convert the given equation into the slope-intercept form [tex]\( y = mx + b \)[/tex], we need to isolate [tex]\( y \)[/tex].
Given:
[tex]\[ y + 3 = -\frac{2}{3}(x - 4) \][/tex]
First, distribute the slope [tex]\(-\frac{2}{3}\)[/tex] on the right-hand side:
[tex]\[ y + 3 = -\frac{2}{3}x + \frac{8}{3} \][/tex]
Next, subtract 3 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{2}{3}x + \frac{8}{3} - 3 \][/tex]
Since 3 can be written as [tex]\(\frac{9}{3}\)[/tex], the equation becomes:
[tex]\[ y = -\frac{2}{3}x + \frac{8}{3} - \frac{9}{3} \][/tex]
Simplify the constants on the right-hand side:
[tex]\[ y = -\frac{2}{3}x - \frac{1}{3} \][/tex]
### Step 2: Identify the Slope and Y-Intercept
From the slope-intercept form [tex]\( y = mx + b \)[/tex], we know:
- The slope [tex]\( m \)[/tex] is [tex]\(-\frac{2}{3} \)[/tex].
- The y-intercept [tex]\( b \)[/tex] is [tex]\(-\frac{1}{3} \)[/tex].
### Step 3: Plot the Graph
Using the identified slope and y-intercept, we can plot the graph:
1. Y-Intercept: Start at the point (0, -[tex]\(\frac{1}{3}\)[/tex]).
2. Using the Slope: The slope [tex]\(-\frac{2}{3}\)[/tex] means from the y-intercept, you go down 2 units for every 3 units you go to the right.
- From (0, -[tex]\(\frac{1}{3}\)[/tex]), move right 3 units to (3, -1).
- From (3, -1), go down 2 units to the next point, which would be (3, -2).
Now you have points and the direction for the line, and you can draw the line through these points.
### Step 4: Compare with Given Graphs
Look at the provided graph options (A, B, C) and find the one that:
- Crosses the y-axis at -[tex]\(\frac{1}{3}\)[/tex].
- Has a downward slope where for every 3 units you move right, you move down 2 units.
By matching these characteristics, you should be able to identify the correct graph.
Since I don't have the visual graphs in front of me, you can use these criteria to match with the right option yourself.
