Alright, let's graph the given line step-by-step:
### Step 1: Rewrite the equation in slope-intercept form
First, we start with the given linear equation:
[tex]\[ y + 1 = -\frac{3}{5} (x - 4) \][/tex]
Our goal is to rewrite this equation in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step 2: Distribute and simplify
Distribute the [tex]\(-\frac{3}{5}\)[/tex] across [tex]\((x - 4)\)[/tex]:
[tex]\[ y + 1 = -\frac{3}{5} x + \frac{12}{5} \][/tex]
Next, subtract 1 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{3}{5} x + \frac{12}{5} - 1 \][/tex]
Change 1 to a fraction with a denominator of 5:
[tex]\[ y = -\frac{3}{5} x + \frac{12}{5} - \frac{5}{5} \][/tex]
Simplify the constants:
[tex]\[ y = -\frac{3}{5} x + \frac{7}{5} \][/tex]
### Step 3: Identify the slope and y-intercept
From the equation [tex]\( y = -\frac{3}{5} x + \frac{7}{5} \)[/tex], we can see:
- The slope [tex]\( m = -\frac{3}{5} \)[/tex]
- The y-intercept [tex]\( b = \frac{7}{5} \)[/tex]
### Step 4: Plot the y-intercept
Locate and plot the y-intercept [tex]\(\frac{7}{5}\)[/tex] on the y-axis. This is approximately 1.4 and can be marked as a point [tex]\((0, \frac{7}{5})\)[/tex].
### Step 5: Use the slope to find another point
The slope [tex]\(-\frac{3}{5}\)[/tex] tells us that for every 5 units we move to the right along the x-axis, the line moves down 3 units.
Starting from the point [tex]\((0, \frac{7}{5})\)[/tex]:
- Move 5 units to the right to [tex]\( (5, \_) \)[/tex]
- Move down 3 units from [tex]\(\frac{7}{5} \text{ to } \frac{7}{5} - 3\)[/tex]
Since [tex]\(\frac{7}{5} - 3 = \frac{7}{5} - \frac{15}{5} = -\frac{8}{5}\)[/tex], the new point is:
[tex]\[ (5, -\frac{8}{5}) \][/tex]
### Step 6: Plot the second point
Plot the second point [tex]\((5, -\frac{8}{5})\)[/tex], which is approximately [tex]\((5, -1.6)\)[/tex] on the graph.
### Step 7: Draw the line
Draw a straight line passing through the points [tex]\((0, \frac{7}{5})\)[/tex] and [tex]\((5, -\frac{8}{5})\)[/tex].
### Step 8: Extend the line
Extend the line in both directions to cover the entire coordinate plane.
### Step 9: Verify additional points for accuracy
Here are values for [tex]\( x \)[/tex] ranging from -10 to 10 with corresponding [tex]\( y \)[/tex] values on the line:
[tex]\[ x \approx -10, y \approx 7.4 \][/tex]
[tex]\[ x \approx -5, y \approx 6 \][/tex]
[tex]\[ x \approx 0, y \approx 1.4 \][/tex]
[tex]\[ x \approx 5, y \approx -1.6 \][/tex]
[tex]\[ x \approx 10, y \approx -4.6 \][/tex]
By following these steps, the line [tex]\( y = -\frac{3}{5} x + \frac{7}{5} \)[/tex] should be accurately drawn on the coordinate plane.
