Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?
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.
To determine which graph shows that the linear system has an infinite number of solutions, we need to find the consistent lines that represent this system.
To begin, let's rewrite the given system of equations in slope-intercept form (y = mx + b) to make it easier to evaluate:
Equation 1: -2x + y = 6
Rewriting it in slope-intercept form: y = 2x + 6
Equation 2: 4x - 2y = -12
Rewriting it in slope-intercept form: y = 2x + 6
We can observe that both equations are in the same form: y = 2x + 6. This means that the lines represented by these equations are identical and would coincide on a graph.
Therefore, the graph that would show that the linear system has an infinite number of solutions would be a graph where the lines of the two equations overlap perfectly, resulting in a single line.
This would be represented by Option A:
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In this graph, the lines of both equations would be indistinguishable, indicating that there are infinitely many points of intersection and thus, an infinite number of solutions for the system of equations.
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 understand the concept of a linear system's solutions.
A linear system consists of two or more linear equations. The solution to a linear system is the unique point or points (x, y) that satisfy all the equations simultaneously. There are three possibilities for a linear system's solution:
1. No solution: The lines representing the equations are parallel and never intersect.
2. Exactly one solution: The lines representing the equations intersect at a single point.
3. Infinite solutions: The lines representing the equations are identical, meaning they overlap and have an infinite number of points in common.
To find the solutions, we can solve the linear system by using one of the following methods:
- Substitution method
- Elimination method
- Graphing method (using a graphing calculator or plotting the equations on a graph)
Now, to identify which graph represents a linear system with an infinite number of solutions, we can plot the given equations on a graph.
For the equation −2x+y=6:
- Rearrange the equation to solve for y:
y = 2x + 6
For the equation 4x−2y=−12:
- Rearrange the equation to solve for y:
y = 2x + 6
Notice that both equations have the same form: y = 2x + 6. This means they represent the same line.
Therefore, the correct graph that shows the linear system has an infinite number of solutions is one where the two lines represented by the equations −2x+y=6 and 4x−2y=−12 are the same line, meaning they completely overlap.
You can check the given graphs to see if any of them depict two identical lines overlapping, indicating an infinite number of solutions for the linear system.