Answer :
To determine which equation could be the second equation in a system that has an infinite number of solutions with the given equation [tex]\(3y = 2x - 9\)[/tex], we need to follow several steps to ensure both equations represent the same line.
### Step 1: Convert the Given Equation to Slope-Intercept Form
The given equation is [tex]\(3y = 2x - 9\)[/tex]. First, we need to convert this into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
1. Divide all terms by 3:
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
In slope-intercept form, the equation is [tex]\(y = \frac{2}{3}x - 3\)[/tex].
### Step 2: Analyze the Given Choices
We will convert each of the given choices to the slope-intercept form to see if it matches the given equation's slope and y-intercept.
#### Choice 1: [tex]\(2y = x - 4.5\)[/tex]
1. Divide all terms by 2:
[tex]\[ y = \frac{1}{2}x - 2.25 \][/tex]
This equation in slope-intercept form is [tex]\(y = \frac{1}{2}x - 2.25\)[/tex].
- Slope: [tex]\(\frac{1}{2}\)[/tex]
- Y-intercept: -2.25
The slope [tex]\(\frac{1}{2}\)[/tex] is not equal to [tex]\(\frac{2}{3}\)[/tex], so this equation does not represent the same line as the given equation [tex]\(y = \frac{2}{3}x - 3\)[/tex]. Therefore, this is not the correct choice.
#### Choice 2: [tex]\(y = \frac{2}{3}x - 3\)[/tex]
The equation is already in slope-intercept form:
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
- Slope: [tex]\(\frac{2}{3}\)[/tex]
- Y-intercept: -3
The slope [tex]\(\frac{2}{3}\)[/tex] and y-intercept -3 are identical to those of the given equation [tex]\(y = \frac{2}{3}x - 3\)[/tex]. Therefore, this equation represents the same line and could be the second equation in the system that has an infinite number of solutions.
### Conclusion
The correct choice for the second equation in the system, which would have an infinite number of solutions with [tex]\(3y = 2x - 9\)[/tex], is:
[tex]\[ \boxed{y = \frac{2}{3}x - 3} \][/tex]
### Step 1: Convert the Given Equation to Slope-Intercept Form
The given equation is [tex]\(3y = 2x - 9\)[/tex]. First, we need to convert this into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
1. Divide all terms by 3:
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
In slope-intercept form, the equation is [tex]\(y = \frac{2}{3}x - 3\)[/tex].
### Step 2: Analyze the Given Choices
We will convert each of the given choices to the slope-intercept form to see if it matches the given equation's slope and y-intercept.
#### Choice 1: [tex]\(2y = x - 4.5\)[/tex]
1. Divide all terms by 2:
[tex]\[ y = \frac{1}{2}x - 2.25 \][/tex]
This equation in slope-intercept form is [tex]\(y = \frac{1}{2}x - 2.25\)[/tex].
- Slope: [tex]\(\frac{1}{2}\)[/tex]
- Y-intercept: -2.25
The slope [tex]\(\frac{1}{2}\)[/tex] is not equal to [tex]\(\frac{2}{3}\)[/tex], so this equation does not represent the same line as the given equation [tex]\(y = \frac{2}{3}x - 3\)[/tex]. Therefore, this is not the correct choice.
#### Choice 2: [tex]\(y = \frac{2}{3}x - 3\)[/tex]
The equation is already in slope-intercept form:
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
- Slope: [tex]\(\frac{2}{3}\)[/tex]
- Y-intercept: -3
The slope [tex]\(\frac{2}{3}\)[/tex] and y-intercept -3 are identical to those of the given equation [tex]\(y = \frac{2}{3}x - 3\)[/tex]. Therefore, this equation represents the same line and could be the second equation in the system that has an infinite number of solutions.
### Conclusion
The correct choice for the second equation in the system, which would have an infinite number of solutions with [tex]\(3y = 2x - 9\)[/tex], is:
[tex]\[ \boxed{y = \frac{2}{3}x - 3} \][/tex]
