1. How many solutions does the system of equations have?
4x=-12y+16 and x+3y=4
A. One
B. Two
C. Infinitely many
D. None
2. How many solutions does the system of equations have?
Y=5x+7 and 3y-15x=18
A. One
B. Two
C. Infinitely many
D. None
3. How many solutions does the system of equations have?
x-2y=6 and 3x-6y=18
A. One
B. Two
C. Infinitely many
D. None
4. How many solutions does the system of equations have?
y-7x=-14 and 7y-49x=-2
A. One
B. Two
C. Infinitely many
D. None
My answers are: A C D A
#1
Note that
x+3y=4
can be rewritten as
4x=-12y+16
so both lines are the same, so C
#2 Y=5x+7
can be written as
3y-15x=21
So, the two lines are parallel: D
#3 x-2y=6
is
3x-6y=18
So, it's the same line: C
#4 y-7x=-14
is
7y-49x=-98
so, D
Looks like you need some further study on these.
I'm not good at these things. Usually I am a fast learner, but this year has been really hard. Thank you!
Yes, it has helped
Answers for test
A
B
A
D
C
B
A
1. The system of equations 4x=-12y+16 and x+3y=4 has infinitely many solutions, just like the number of options in a buffet. So the answer is C. Infinitely many.
2. The system of equations Y=5x+7 and 3y-15x=18 is as confused as a chicken trying to dance ballet. It has no common ground, so the answer is D. None.
3. Ah, the system of equations x-2y=6 and 3x-6y=18. These equations are like twins separated at birth, always matching each other. It has only one solution, like finding your missing sock in the dryer. So the answer is A. One.
4. Now we have the system of equations y-7x=-14 and 7y-49x=-2. These equations are as inconsistent as mismatched socks. They have no solution, just like trying to fit a square peg in a round hole. So the answer is D. None.
To find the number of solutions for each system of equations, we need to analyze the coefficients and constants in the equations. Here's how you can determine the answer for each question:
1. For the given system:
4x = -12y + 16
x + 3y = 4
To find the number of solutions, we can use the concept of slope-intercept form. Rearrange the second equation to isolate y and write it in slope-intercept form:
x + 3y = 4
3y = -x + 4
y = (-1/3)x + 4/3
Comparing the coefficients of x in both equations, we see that the slopes are different (-1 and -1/3). Since the slopes don't match, the lines representing the equations are not parallel and will intersect at a single point. Therefore, there is only one solution. So the answer is A.
2. For the given system:
y = 5x + 7
3y - 15x = 18
Again, let's rewrite the second equation in slope-intercept form by isolating y:
3y - 15x = 18
3y = 15x + 18
y = 5x + 6
Comparing the coefficients of x in both equations, we see that the slopes (5 and 5) are the same. Since the slopes match and the y-intercepts (7 and 6) are different, the lines representing the equations are parallel and will never intersect. Therefore, there are no solutions. So the answer is D.
3. For the given system:
x - 2y = 6
3x - 6y = 18
Simplifying the second equation, we get:
3x - 6y = 18
x - 2y = 6
Notice that both equations are essentially the same. This means that the second equation is a multiple of the first equation. Therefore, there are infinitely many solutions, and the lines representing the equations are coincident. So the answer is C.
4. For the given system:
y - 7x = -14
7y - 49x = -2
Dividing the second equation by 7, we get:
y - 7x = -14
y - 7x = -2/7
Comparing both equations, we can see that they are parallel lines with the same slope (-7). However, the y-intercepts (-14 and -2/7) are different. Since the lines are parallel and not coincident, they will never intersect, resulting in no common solution. Therefore, the answer is D.
So, the correct answers are:
1. A (One)
2. D (None)
3. C (Infinitely many)
4. D (None)
y=
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y=
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