Sure, let's determine the correct equation for the given line in slope-intercept form.
1. The slope-intercept form of a linear equation is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. We need to identify the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) from the provided options.
3. Let's examine each option to find the correct match:
- Option 1: [tex]\( y = -\frac{5}{3} x - 1 \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\(-\frac{5}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\(-1\)[/tex]
- Option 2: [tex]\( y = \frac{5}{3} x + 1 \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\(\frac{5}{3}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\(1\)[/tex]
- Option 3: [tex]\( y = \frac{3}{5} x + 1 \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\(\frac{3}{5}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\(1\)[/tex]
- Option 4: [tex]\( y = -\frac{3}{5} x - 1 \)[/tex]
- Slope ([tex]\( m \)[/tex]): [tex]\(-\frac{3}{5}\)[/tex]
- Y-intercept ([tex]\( b \)[/tex]): [tex]\(-1\)[/tex]
4. We want the equation that accurately reflects the given slope and y-intercept.
- Given slope: [tex]\(-\frac{3}{5}\)[/tex]
- Given y-intercept: [tex]\(-1\)[/tex]
5. Comparing these values with the options:
- Option 1 has the incorrect slope.
- Option 2 has both the incorrect slope and y-intercept.
- Option 3 has the incorrect slope and y-intercept.
- Option 4 has the correct slope ([tex]\(-\frac{3}{5}\)[/tex]) and the correct y-intercept ([tex]\(-1\)[/tex]).
Therefore, the correct equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{3}{5} x - 1 \][/tex]
So, the answer is:
[tex]\[ y = -\frac{3}{5} x - 1 \][/tex]
