Question
If a system of equations has an infinite number of solutions, what is it called? Choose the best answer. Enter the number associated with that answer for your response. IT IS NOT 2
inconsistent and dependent system of equations
consistent system of equations
dependent system of equations
consistent and dependent system of equations
inconsistent and dependent system of equations
consistent system of equations
dependent system of equations
consistent and dependent system of equations
Answers
Answer
Solve the given system of linear equations by inspection. Enter 1 for one solution, 2 for no solution, or 3 for infinitely many solutions.
2x−7y=4
2x−7y=−4
2x−7y=4
2x−7y=−4
Answer
Solve the given system of linear equations by inspection. Enter 1 for one solution, 2 for no solution, or 3 for infinitely many solutions.
y=−3x+5
2y=−6x+10
y=−3x+5
2y=−6x+10
Answer
Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.
(1 point)
Responses
The statement is false, because the lines have an intersection point.
The statement is false, because the lines have an intersection point.
The statement is true, because the lines are parallel.
The statement is true, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is true, because the lines have an intersection point.
(1 point)
Responses
The statement is false, because the lines have an intersection point.
The statement is false, because the lines have an intersection point.
The statement is true, because the lines are parallel.
The statement is true, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is false, because the lines are parallel.
The statement is true, because the lines have an intersection point.
Answer
Does the graph show the system of equations x+y=2 and −x+y=7 ? Should the lines for the system be parallel?
(1 point)
Responses
The graph of x+y=2 is incorrect. The lines should intersect.
The graph of x plus y equals 2 is incorrect. The lines should intersect.
Both graphs are correct. The lines should be parallel.
Both graphs are correct. The lines should be parallel.
The graph of −x+y=7 is incorrect. The lines should be parallel.
The graph of negative x plus y equals 7 is incorrect. The lines should be parallel.
The graph of −x+y=7 is incorrect. The lines should intersect.
(1 point)
Responses
The graph of x+y=2 is incorrect. The lines should intersect.
The graph of x plus y equals 2 is incorrect. The lines should intersect.
Both graphs are correct. The lines should be parallel.
Both graphs are correct. The lines should be parallel.
The graph of −x+y=7 is incorrect. The lines should be parallel.
The graph of negative x plus y equals 7 is incorrect. The lines should be parallel.
The graph of −x+y=7 is incorrect. The lines should intersect.
Answer
Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?
Answered by
GPT-4o mini
4
Answered by
GPT-4o mini
The two equations are:
1. \( 2x - 7y = 4 \)
2. \( 2x - 7y = -4 \)
These equations have the same left-hand side but different right-hand sides. This means that the lines represented by these equations are parallel and will never intersect.
Therefore, the system of equations has **no solution.**
The correct response is **2**.
1. \( 2x - 7y = 4 \)
2. \( 2x - 7y = -4 \)
These equations have the same left-hand side but different right-hand sides. This means that the lines represented by these equations are parallel and will never intersect.
Therefore, the system of equations has **no solution.**
The correct response is **2**.
Answered by
GPT-4o mini
Let's analyze the two equations:
1. \( y = -3x + 5 \)
2. \( 2y = -6x + 10 \)
We can simplify the second equation by dividing everything by 2:
\[
y = -3x + 5
\]
Now both equations are identical. Since they represent the same line, they have infinitely many solutions.
Thus, the correct response is **3**.
1. \( y = -3x + 5 \)
2. \( 2y = -6x + 10 \)
We can simplify the second equation by dividing everything by 2:
\[
y = -3x + 5
\]
Now both equations are identical. Since they represent the same line, they have infinitely many solutions.
Thus, the correct response is **3**.
Answered by
GPT-4o mini
Let's analyze the given equations:
1. \( y = -x + 3 \) (This is in slope-intercept form)
2. \( x + y = 8 \) (We can rearrange this to slope-intercept form: \( y = -x + 8 \))
Now, comparing the two equations:
1. The first equation, \( y = -x + 3 \), has a slope of -1 and a y-intercept of 3.
2. The second equation, rearranged to \( y = -x + 8 \), also has a slope of -1 but a y-intercept of 8.
Since both lines have the same slope (-1) but different y-intercepts (3 and 8), they are parallel and will never intersect. Therefore, there is no solution to the system.
The correct response is:
**The statement is true, because the lines are parallel.**
1. \( y = -x + 3 \) (This is in slope-intercept form)
2. \( x + y = 8 \) (We can rearrange this to slope-intercept form: \( y = -x + 8 \))
Now, comparing the two equations:
1. The first equation, \( y = -x + 3 \), has a slope of -1 and a y-intercept of 3.
2. The second equation, rearranged to \( y = -x + 8 \), also has a slope of -1 but a y-intercept of 8.
Since both lines have the same slope (-1) but different y-intercepts (3 and 8), they are parallel and will never intersect. Therefore, there is no solution to the system.
The correct response is:
**The statement is true, because the lines are parallel.**
Answered by
GPT-4o mini
Let's analyze the two equations:
1. \( x + y = 2 \)
2. \( -x + y = 7 \)
First, we can rearrange them to slope-intercept form \( y = mx + b \):
1. For the first equation \( x + y = 2 \):
\[
y = -x + 2
\]
(Slope = -1, Y-intercept = 2)
2. For the second equation \( -x + y = 7 \):
\[
y = x + 7
\]
(Slope = 1, Y-intercept = 7)
Now, compare the slopes:
- The slope of the first line is -1.
- The slope of the second line is 1.
Since the slopes are different, the lines are **not parallel** and will intersect at some point.
Therefore, the correct response is:
**The graph of \( -x + y = 7 \) is incorrect. The lines should intersect.**
1. \( x + y = 2 \)
2. \( -x + y = 7 \)
First, we can rearrange them to slope-intercept form \( y = mx + b \):
1. For the first equation \( x + y = 2 \):
\[
y = -x + 2
\]
(Slope = -1, Y-intercept = 2)
2. For the second equation \( -x + y = 7 \):
\[
y = x + 7
\]
(Slope = 1, Y-intercept = 7)
Now, compare the slopes:
- The slope of the first line is -1.
- The slope of the second line is 1.
Since the slopes are different, the lines are **not parallel** and will intersect at some point.
Therefore, the correct response is:
**The graph of \( -x + y = 7 \) is incorrect. The lines should intersect.**
Answered by
GPT-4o mini
To determine if the linear system has an infinite number of solutions, we first need to analyze the equations:
1. **Equation 1**: \( -2x + y = 6 \)
2. **Equation 2**: \( 4x - 2y = -12 \)
To see if the two equations represent the same line (which would indicate an infinite number of solutions), we can manipulate the second equation.
First, we can rewrite the second equation:
\[
4x - 2y = -12
\]
We can divide the entire equation by 2 to simplify it:
\[
2x - y = -6
\]
Next, we can rearrange this into slope-intercept form (y = mx + b):
\[
y = 2x + 6
\]
Now, let's review the first equation:
Starting from \( -2x + y = 6 \):
We can rewrite it in slope-intercept form:
\[
y = 2x + 6
\]
We can see that both equations rewrite to the same equation:
\[
y = 2x + 6
\]
Since both equations represent the same line, we can conclude that the system has an **infinite number of solutions**.
To find the graph corresponding to this conclusion, look for a graph where both lines overlap completely (they are the exact same line).
If you have specific graphs to choose from, please share them so I can help identify which graph shows an infinite number of solutions. Since they represent the same line, the correct graph would depict one line (and not two distinct lines).
1. **Equation 1**: \( -2x + y = 6 \)
2. **Equation 2**: \( 4x - 2y = -12 \)
To see if the two equations represent the same line (which would indicate an infinite number of solutions), we can manipulate the second equation.
First, we can rewrite the second equation:
\[
4x - 2y = -12
\]
We can divide the entire equation by 2 to simplify it:
\[
2x - y = -6
\]
Next, we can rearrange this into slope-intercept form (y = mx + b):
\[
y = 2x + 6
\]
Now, let's review the first equation:
Starting from \( -2x + y = 6 \):
We can rewrite it in slope-intercept form:
\[
y = 2x + 6
\]
We can see that both equations rewrite to the same equation:
\[
y = 2x + 6
\]
Since both equations represent the same line, we can conclude that the system has an **infinite number of solutions**.
To find the graph corresponding to this conclusion, look for a graph where both lines overlap completely (they are the exact same line).
If you have specific graphs to choose from, please share them so I can help identify which graph shows an infinite number of solutions. Since they represent the same line, the correct graph would depict one line (and not two distinct lines).