Determine the number of solutions to this system of linear equations:
2x+y=6
4x+2y=8(1 point)
Responses
1
1
0
0
infinite
infinite
2
11 answers
1
Question
Which of the graphs with no solution correctly shows the system of equations y=2x+4 and −2x+y=8?(1 point)
Responses
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
Which of the graphs with no solution correctly shows the system of equations y=2x+4 and −2x+y=8?(1 point)
Responses
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 2 comma 0 right parenthesis and left parenthesis 0 comma 4 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 8 right parenthesis.
A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants ranges from negative 10 to 10 in unit increments on both the x and y axes. A solid line and a dashed line with arrows at both the ends are drawn parallel to each other on the graph. The solid line passes through left parenthesis 0 comma 4 right parenthesis and left parenthesis negative 2 comma 0 right parenthesis. The dashed lines pass through left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 8 right parenthesis.
The correct graph with no solution for the system of equations y=2x+4 and −2x+y=8 is the first one described, where a solid line and a dashed line are drawn parallel to each other and do not intersect. This indicates that there is no solution to the system of equations.
Question
Which of these graphs shows that the linear system 3x+2y=12 and 6x+4y=24 has an infinite number of solutions?(1 point)
Responses
A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.
Which of these graphs shows that the linear system 3x+2y=12 and 6x+4y=24 has an infinite number of solutions?(1 point)
Responses
A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.
The correct graph that shows the linear system 3x+2y=12 and 6x+4y=24 has an infinite number of solutions is the second one described with two parallel lines. This situation indicates that the two equations represent the same line, meaning they have an infinite number of points of intersection or solutions.
E
I'm sorry for the mistake in my previous response. The correct graph that shows the linear system 3x+2y=12 and 6x+4y=24 has an infinite number of solutions is not options provided. The correct graph would be two overlapping lines, where one equation is a multiple of the other. This overlapping or coinciding lines would indicate an infinite number of solutions.
Solve the system of equations.
4x−8y=−36
−6x+y=32(1 point)
(
,
)
4x−8y=−36
−6x+y=32(1 point)
(
,
)
To solve the system of equations:
4x - 8y = -36
-6x + y = 32
We can use the method of substitution or elimination. Let's use the elimination method to solve:
1. Multiply the second equation by 8 so that the coefficient of y will become -8, allowing easy elimination of y.
-6x + y = 32
8(-6x + y) = 8(32)
-48x + 8y = 256
Now we have:
4x - 8y = -36
-48x + 8y = 256
2. Add the equations to eliminate y:
(4x - 8y) + (-48x + 8y) = -36 + 256
-44x = 220
x = -220 / 44
x = -5
3. Substitute x back into one of the original equations to solve for y. Let's use the first equation:
4(-5) - 8y = -36
-20 - 8y = -36
-8y = -16
y = 2
Therefore, the solution to the system of equations 4x - 8y = -36 and -6x + y = 32 is (x, y) = (-5, 2).
4x - 8y = -36
-6x + y = 32
We can use the method of substitution or elimination. Let's use the elimination method to solve:
1. Multiply the second equation by 8 so that the coefficient of y will become -8, allowing easy elimination of y.
-6x + y = 32
8(-6x + y) = 8(32)
-48x + 8y = 256
Now we have:
4x - 8y = -36
-48x + 8y = 256
2. Add the equations to eliminate y:
(4x - 8y) + (-48x + 8y) = -36 + 256
-44x = 220
x = -220 / 44
x = -5
3. Substitute x back into one of the original equations to solve for y. Let's use the first equation:
4(-5) - 8y = -36
-20 - 8y = -36
-8y = -16
y = 2
Therefore, the solution to the system of equations 4x - 8y = -36 and -6x + y = 32 is (x, y) = (-5, 2).
Question
Solve the system of equations.
6x−3y=−6
9x−3y=−18 (1 point)
(
,
)
Solve the system of equations.
6x−3y=−6
9x−3y=−18 (1 point)
(
,
)
To solve the system of equations:
6x - 3y = -6
9x - 3y = -18
We can see that the two equations have the same coefficients for x and y, which implies that they represent the same line. Therefore, they have infinitely many solutions.
The solution to this system of equations is all points that lie on the line represented by the equation 6x - 3y = -6 or 9x - 3y = -18. This system does not have a unique solution but rather an infinite number of solutions that lie on the line represented by the given equations.
6x - 3y = -6
9x - 3y = -18
We can see that the two equations have the same coefficients for x and y, which implies that they represent the same line. Therefore, they have infinitely many solutions.
The solution to this system of equations is all points that lie on the line represented by the equation 6x - 3y = -6 or 9x - 3y = -18. This system does not have a unique solution but rather an infinite number of solutions that lie on the line represented by the given equations.