Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?(1 point)
To determine if a linear system has an infinite number of solutions, we need to check if the two equations are equivalent or if they represent the same line.
Let's rearrange the first equation −2x + y = 6:
y = 2x + 6
Now, let's rearrange the second equation 4x − 2y = −12:
-2y = -4x - 12
y = 2x + 6
Both equations can be simplified to y = 2x + 6, which means the two equations represent the same line.
Therefore, the graphs of the linear system −2x+y=6 and 4x−2y=−12 would show a set of coinciding lines or one line on top of another, indicating an infinite number of solutions.
This can be represented by the graph:
Graph:
y = 2x + 6
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 examine the slopes of the two lines.
The given system can be rewritten in slope-intercept form as follows:
-2x+y=6
y=2x+6
4x-2y=-12
-2y = -4x - 12
y = 2x + 6
By comparing the slopes of the two lines, we can determine if they are parallel or coincident:
The slope of the first line is 2, and the slope of the second line is also 2.
Since the slopes are equal, it indicates that the lines are coincident or overlapping, meaning there are infinite solutions.
Therefore, the graph that shows an infinite number of solutions for the given system is the one where the lines are coincident or overlapping.
To determine which graph shows that the given linear system has an infinite number of solutions, we need to examine the slopes and y-intercepts of the two equations.
The linear system can be rewritten in slope-intercept form (y = mx + b) to easily determine the slope and y-intercept of each equation.
For the first equation −2x + y = 6, we need to isolate the y variable:
y = 2x + 6
The slope of this equation is 2, indicating that the line will be upward sloping.
For the second equation 4x − 2y = −12:
-2y = -4x - 12
y = 2x + 6
This equation represents the same line as the first equation, with identical slopes and y-intercepts. Therefore, the two equations are equivalent and represent the same line.
To have an infinite number of solutions for a system of linear equations, the equations need to be dependent, meaning they represent the same line. Therefore, any graph that shows both equations as overlapping or coinciding lines will demonstrate an infinite number of solutions.
Select the graph where the lines of both equations overlap or coincide, indicating an infinite number of solutions.