Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 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. Two intersecting lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 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 upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 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 upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.
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 upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.
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. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 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. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis.
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. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 3 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. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 3 comma 0 right parenthesis.
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The linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions when the two equations represent the same line. The graph that shows this is the first one described, where a solid upward slanting line passes through the points (-3, 0) and (0, 6), and a dotted upward slanting line passes through the points (0, 6) and (1, 7).
Is it a b c or d
The correct answer is A.
The graph that shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions is A.
In the graph, there are two intersecting lines: a solid upward slanting line passing through the points (-3, 0) and (0, 6), and a dotted upward slanting line passing through the points (0, 6) and (1, 7). These lines intersect at the point (0, 6). Since the lines intersect at a single point, this indicates that there is a unique solution to the system of equations, and therefore, it does not have an infinite number of solutions.
To determine which graph shows that the linear system -2x + y = 6 and 4x - 2y = -12 has an infinite number of solutions, we need to analyze the slopes and y-intercepts of the equations.
The first equation, -2x + y = 6, can be rewritten as y = 2x + 6. From this form, we can identify that the slope is 2, and the y-intercept is 6.
The second equation, 4x - 2y = -12, can be rewritten as 2y = 4x + 12, then y = 2x + 6. From this form, we can also identify that the slope is 2, and the y-intercept is 6.
Now let's analyze the options:
A) This graph shows two intersecting lines. The first line passes through the points (-3, 0) and (0, 6), while the second line passes through (0, 6) and (1, 7). Since these lines intersect at the point (0, 6) and have the same slope and y-intercept, this graph represents a linear system with infinite solutions.
B) This graph shows two parallel lines. The first line passes through the points (-3, 0) and (0, 6), while the second line passes through the origin (0,0). Since these lines have different slopes (one is undefined, and the other is 2) and different y-intercepts, this graph represents a linear system with no solutions.
C) This graph shows only one line passing through the points (-3, 0) and (0, 6). Since there is only one line, there is only one unique solution.
D) This graph shows one line passing through the points (0, -6) and (3, 0). Since there is only one line, there is only one unique solution.
Therefore, the correct answer is option A) that shows two intersecting lines passing through the points (-3, 0), (0, 6), and (1, 7), indicating an infinite number of solutions for the linear system.