Answer :
To determine which of the given equations could be the other equation in the system that has an infinite number of solutions with [tex]\(3y = 2x - 9\)[/tex], we need to convert the given equation and possible equations into their slope-intercept form, [tex]\(y = mx + b\)[/tex].
First, let's rewrite the given equation:
[tex]\[ 3y = 2x - 9 \][/tex]
To convert to slope-intercept form ([tex]\(y = mx + b\)[/tex]), divide every term by 3:
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
Now, the slope-intercept form of this equation is [tex]\( y = \frac{2}{3}x - 3 \)[/tex]. The slope ([tex]\(m\)[/tex]) here is [tex]\(\frac{2}{3}\)[/tex] and the y-intercept ([tex]\(b\)[/tex]) is [tex]\(-3\)[/tex].
For two linear equations to have an infinite number of solutions, they must be identical, meaning they must have the same slope and y-intercept.
Now we will check each given equation to see if any match our converted form:
### Option 1: [tex]\(2y = x - 4.5\)[/tex]
Convert to [tex]\(y = mx + b\)[/tex]:
[tex]\[ 2y = x - 4.5 \][/tex]
Divide every term by 2:
[tex]\[ y = \frac{1}{2}x - 2.25 \][/tex]
The slope is [tex]\(\frac{1}{2}\)[/tex], which does not match [tex]\(\frac{2}{3}\)[/tex]. So this equation does not have an infinite number of solutions with the given equation.
### Option 2: [tex]\(y = \frac{2}{3} x - 3\)[/tex]
This is already in slope-intercept form. The slope is [tex]\(\frac{2}{3}\)[/tex] and the y-intercept is [tex]\(-3\)[/tex].
This exactly matches our converted equation. Hence, this equation has an infinite number of solutions with [tex]\(3y = 2x - 9\)[/tex].
### Option 3: [tex]\(6y = 6x - 27\)[/tex]
Convert to [tex]\(y = mx + b\)[/tex]:
[tex]\[ 6y = 6x - 27 \][/tex]
Divide every term by 6:
[tex]\[ y = x - 4.5 \][/tex]
The slope is [tex]\(1\)[/tex], which does not match [tex]\(\frac{2}{3}\)[/tex]. So this equation does not have an infinite number of solutions with the given equation.
### Option 4: [tex]\(y = \frac{3}{2} x - 4.5\)[/tex]
This is already in slope-intercept form. The slope is [tex]\(\frac{3}{2}\)[/tex], which does not match [tex]\(\frac{2}{3}\)[/tex].
Thus, this equation does not have an infinite number of solutions with the given equation.
To summarize, the only equation that could be the other equation in the system to yield an infinite number of solutions alongside [tex]\(3y = 2x - 9\)[/tex] is:
[tex]\[ y = \frac{2}{3} x - 3 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{2} \][/tex]
First, let's rewrite the given equation:
[tex]\[ 3y = 2x - 9 \][/tex]
To convert to slope-intercept form ([tex]\(y = mx + b\)[/tex]), divide every term by 3:
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
Now, the slope-intercept form of this equation is [tex]\( y = \frac{2}{3}x - 3 \)[/tex]. The slope ([tex]\(m\)[/tex]) here is [tex]\(\frac{2}{3}\)[/tex] and the y-intercept ([tex]\(b\)[/tex]) is [tex]\(-3\)[/tex].
For two linear equations to have an infinite number of solutions, they must be identical, meaning they must have the same slope and y-intercept.
Now we will check each given equation to see if any match our converted form:
### Option 1: [tex]\(2y = x - 4.5\)[/tex]
Convert to [tex]\(y = mx + b\)[/tex]:
[tex]\[ 2y = x - 4.5 \][/tex]
Divide every term by 2:
[tex]\[ y = \frac{1}{2}x - 2.25 \][/tex]
The slope is [tex]\(\frac{1}{2}\)[/tex], which does not match [tex]\(\frac{2}{3}\)[/tex]. So this equation does not have an infinite number of solutions with the given equation.
### Option 2: [tex]\(y = \frac{2}{3} x - 3\)[/tex]
This is already in slope-intercept form. The slope is [tex]\(\frac{2}{3}\)[/tex] and the y-intercept is [tex]\(-3\)[/tex].
This exactly matches our converted equation. Hence, this equation has an infinite number of solutions with [tex]\(3y = 2x - 9\)[/tex].
### Option 3: [tex]\(6y = 6x - 27\)[/tex]
Convert to [tex]\(y = mx + b\)[/tex]:
[tex]\[ 6y = 6x - 27 \][/tex]
Divide every term by 6:
[tex]\[ y = x - 4.5 \][/tex]
The slope is [tex]\(1\)[/tex], which does not match [tex]\(\frac{2}{3}\)[/tex]. So this equation does not have an infinite number of solutions with the given equation.
### Option 4: [tex]\(y = \frac{3}{2} x - 4.5\)[/tex]
This is already in slope-intercept form. The slope is [tex]\(\frac{3}{2}\)[/tex], which does not match [tex]\(\frac{2}{3}\)[/tex].
Thus, this equation does not have an infinite number of solutions with the given equation.
To summarize, the only equation that could be the other equation in the system to yield an infinite number of solutions alongside [tex]\(3y = 2x - 9\)[/tex] is:
[tex]\[ y = \frac{2}{3} x - 3 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{2} \][/tex]
