What do you notice about these graphs? What does it mean for a system to have
infinite solutions? What do the graphs of a system with infinite solutions look like?
Explain what it means graphically to have infinite solutions.
Reponses area
Plug a few points into the original equations and check to show that the system of
equations has more than one solution. Explain your work.
Reponses area
Based on the given question, there are no graphs or equations provided to observe. Therefore, it is not possible to notice any specific characteristics about the graphs.
When a system of equations has infinite solutions, it means that every point on the graph of one equation is also on the graph of the other equation. In other words, the two lines representing the equations are the same line. This indicates that the two equations are dependent and represent the same relationship between variables.
Graphically, this can be represented by two lines that coincide and completely overlap each other. This means that all points on one line are also on the other line, resulting in an infinite number of overlapping solutions.
To demonstrate that a system of equations has infinite solutions by plugging points, we would choose any set of coordinates on the graph of one equation and check if they satisfy the other equation as well. If they do, it confirms that these equations have more than one solution.
1. In the question, it is mentioned that there are multiple graphs. However, as an AI, I do not have access to any specific graphs. Therefore, I cannot provide any specific insights about the graphs you are referring to.
2. A system of equations is said to have infinite solutions when there are infinitely many different combinations of values for the variables that satisfy all the equations in the system. In other words, there is no unique solution but rather an infinite number of solutions that satisfy the system.
3. Graphically, a system with infinite solutions usually consists of two or more lines that are coincident or overlap perfectly. This means that all the lines in the system are superimposed on each other, and any point on the common line(s) will satisfy all the equations in the system.
4. To demonstrate that a system of equations has more than one solution, you can choose a set of values, substitute them into the equations, and verify that they satisfy all the equations. If the values work for all the equations, it implies there are multiple solutions.
For example, let's consider a system of equations:
Equation 1: 2x + y = 5
Equation 2: 4x + 2y = 10
Let's try substituting the values x = 1 and y = 3 into both equations.
Equation 1: 2(1) + 3 = 5 (which is true)
Equation 2: 4(1) + 2(3) = 10 (which is also true)
The same set of values satisfies both equations, indicating that there is more than one solution to the system.