Sure, let's find the slope and y-intercept for the line passing through the points provided in the table.
The points given are:
- [tex]\( (4, 1) \)[/tex]
- [tex]\( (0, 8) \)[/tex]
### Step-by-Step Solution:
1. Finding the Slope ([tex]\( m \)[/tex]):
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using our points [tex]\( (4, 1) \)[/tex] and [tex]\( (0, 8) \)[/tex]:
[tex]\[
m = \frac{8 - 1}{0 - 4} = \frac{7}{-4} = -\frac{7}{4}
\][/tex]
2. Finding the Y-Intercept ([tex]\( b \)[/tex]):
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( b \)[/tex] is the y-intercept. To find [tex]\( b \)[/tex], we can use one of the points and the slope we just calculated. Let's use the point [tex]\( (4, 1) \)[/tex]:
[tex]\[
y = -\frac{7}{4}x + b
\][/tex]
Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[
1 = -\frac{7}{4}(4) + b
\][/tex]
[tex]\[
1 = -7 + b
\][/tex]
Now solve for [tex]\( b \)[/tex]:
[tex]\[
b = 1 + 7 = 8
\][/tex]
3. Writing the Equation:
Now that we have both the slope and the y-intercept, we can write the equation of the line in slope-intercept form:
[tex]\[
y = -\frac{7}{4}x + 8
\][/tex]
Out of the given options, the correct equation of the line is:
[tex]\[
y = -\frac{7}{4} x + 8
\][/tex]
