To determine if a system of linear equations has an infinite number of solutions, we need to check if the two lines representing the equations in the system are coincident or overlapping. This occurs when the two equations represent the same line.
We can rewrite the given equations in slope-intercept form (y = mx + b) to better analyze them:
For the equation −2x + y = 6:
Add 2x to both sides: y = 2x + 6
For the equation 4x − 2y = −12:
Divide both sides by 2: 2x - y = -6
Rearrange terms: y = 2x + 6
Comparing the rewritten equations, we can see that they are identical: y = 2x + 6. This means that the system of equations has infinitely many solutions.
However, since we are given graphs as options, we can determine this visually by analyzing the slope and y-intercept of the lines represented in the graphs:
The graphs showing lines with the same slope (2) and y-intercept (6) indicate that the system has an infinite number of solutions. Therefore, the answer is either A or E, depending on the precise coordinates of the points on the graphs.
Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?
1 answer