Which of these graphs show that the linear system -2x+y=6 and 4x-2y=-12 has an infinite number of solutions?
To determine if the linear system has an infinite number of solutions, we need to graph the two equations and see if they intersect at multiple points or if the lines are coincident (i.e., the same line).
The given system of equations is:
-2x + y = 6 ----(1)
4x - 2y = -12 ----(2)
We can rewrite equation (1) in terms of y:
y = 2x + 6
Now let's graph the two equations to determine their relationship.
Graph 1:
Since equation (1) is in slope-intercept form (y = mx + b), we can easily identify that the slope is 2 and the y-intercept is 6.
Starting at the y-intercept at 6, we move upward 2 units and then one unit to the right to find a second point on the line. We can continue this process to find additional points. Once we draw the line through these points, we have the graph of equation (1).
Graph 2:
Rewriting equation (2) in terms of y, we have:
2y = 4x + 12
y = 2x + 6
When we compare this equation to equation (1), we observe that they are exactly the same. This means that the two equations represent the same line and therefore have an infinite number of solutions.
Therefore, in order to determine which graph shows the infinitely many solutions, you would select the graph where both equations are represented by the same line. In this case, it would be Graph 2.