Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?(1 point)
The graph that shows an infinite number of solutions for the linear system −2x+y=6 and 4x−2y=−12 is the graph in which the two lines are coincident or overlapping, forming a single line.
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 equations and their slopes.
The equations of the linear system are:
-2x + y = 6 (equation 1)
4x - 2y = -12 (equation 2)
To find the slope of equation 1, we can rewrite it in slope-intercept form (y = mx + b) by isolating y:
y = 2x + 6
From this equation, we can see that the slope of equation 1 is 2.
Now, let's find the slope of equation 2:
4x - 2y = -12
-2y = -4x - 12
y = 2x + 6
From this equation, we can see that the slope of equation 2 is also 2.
Since the slopes of both equations are the same, this indicates that the lines are parallel, and they will never intersect.
Therefore, the answer is: None of the graphs will show that the linear system has an infinite number of solutions.
To determine which graph shows that the linear system has an infinite number of solutions, we need to look for a condition that indicates the lines are coincident or overlapping. In other words, we need to find the graphs where the two equations represent the same line.
To do this, we can rearrange each equation into slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
1) −2x + y = 6:
Adding 2x to both sides, we get:
y = 2x + 6
2) 4x − 2y = −12:
Rearranging the equation, we get:
-2y = -4x - 12
Divide through by -2:
y = 2x + 6
Comparing the two equations, we see that they are identical: y = 2x + 6. This means that the two lines are coincident, or they intersect at every point. Therefore, there are infinitely many solutions to the system of equations.
The correct graph that represents this is the one where the two lines completely overlap or coincide.