Answer :
To determine which equation of the line in slope-intercept form matches the given slope and y-intercept, we need to recall the slope-intercept form of the equation of a line, which is:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept, the point where the line crosses the y-axis.
We are given:
- The slope [tex]\( m = \frac{1}{3} \)[/tex]
- The y-intercept [tex]\( b = -4 \)[/tex]
Substituting these values into the slope-intercept form, we get:
[tex]\[ y = \left( \frac{1}{3} \right)x + (-4) \][/tex]
or, simplifying,
[tex]\[ y = \frac{1}{3}x - 4 \][/tex]
Now, let's match this with the given options:
A. [tex]\( x=\frac{1}{3} y-4 \)[/tex]
B. [tex]\( y=\frac{1}{3} x+4 \)[/tex]
C. [tex]\( y=\frac{1}{3} x-4 \)[/tex]
D. [tex]\( y=-\frac{1}{3} x-4 \)[/tex]
The correct equation that matches [tex]\( y = \frac{1}{3}x - 4 \)[/tex] is:
C. [tex]\( y = \frac{1}{3} x - 4 \)[/tex]
Therefore, the correct answer is option 3.
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept, the point where the line crosses the y-axis.
We are given:
- The slope [tex]\( m = \frac{1}{3} \)[/tex]
- The y-intercept [tex]\( b = -4 \)[/tex]
Substituting these values into the slope-intercept form, we get:
[tex]\[ y = \left( \frac{1}{3} \right)x + (-4) \][/tex]
or, simplifying,
[tex]\[ y = \frac{1}{3}x - 4 \][/tex]
Now, let's match this with the given options:
A. [tex]\( x=\frac{1}{3} y-4 \)[/tex]
B. [tex]\( y=\frac{1}{3} x+4 \)[/tex]
C. [tex]\( y=\frac{1}{3} x-4 \)[/tex]
D. [tex]\( y=-\frac{1}{3} x-4 \)[/tex]
The correct equation that matches [tex]\( y = \frac{1}{3}x - 4 \)[/tex] is:
C. [tex]\( y = \frac{1}{3} x - 4 \)[/tex]
Therefore, the correct answer is option 3.
