Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
