To determine the correct equation for the given line in slope-intercept form, let's analyze the given choices.
In slope-intercept form, an equation of a line is written as:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
Given the equations:
1. [tex]\( y = -\frac{5}{3} x - 1 \)[/tex]
2. [tex]\( y = \frac{5}{3} x + 1 \)[/tex]
3. [tex]\( y = \frac{3}{5} x + 1 \)[/tex]
4. [tex]\( y = -\frac{3}{5} x - 1 \)[/tex]
We need to identify the equation with the correct slope and y-intercept. Let’s examine the slope and y-intercept of each equation:
1. Equation 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]
2. Equation 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]
3. Equation 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]
4. Equation 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]
From these observations:
- Equation 1 has a slope of [tex]\(-\frac{5}{3}\)[/tex] and y-intercept of [tex]\(-1\)[/tex].
- Equation 2 has a slope of [tex]\(\frac{5}{3}\)[/tex] and y-intercept of [tex]\(1\)[/tex].
- Equation 3 has a slope of [tex]\(\frac{3}{5}\)[/tex] and y-intercept of [tex]\(1\)[/tex].
- Equation 4 has a slope of [tex]\(-\frac{3}{5}\)[/tex] and y-intercept of [tex]\(-1\)[/tex].
The correct equation depends on a specific requirement for both the slope and y-intercept. Given these four options, we identify the correct parameters:
- The slope is [tex]\(-\frac{3}{5}\)[/tex].
- The y-intercept is [tex]\(-1\)[/tex].
Therefore, the correct equation is:
[tex]\[ y = -\frac{3}{5} x - 1 \][/tex]
This corresponds to the fourth choice. So, the correct equation for the given line is:
[tex]\[ \boxed{y = -\frac{3}{5} x - 1} \][/tex]
