The following system of equations has infinitely many solutions. Explain what it means for a system of equations to have infinitely many solutions. Use calculations and/or examples to justify your answer.
−x−y=3
5x+5y=−15
To determine if a system of equations has infinitely many solutions, we need to check if the equations are dependent. If the equations in the system are dependent, it means that one equation can be obtained by performing operations on the other equation, resulting in the same line or equations with the same slope and y-intercept.
Let's start by manipulating the first equation:
−x−y=3
Multiply by -1: x + y = -3
Now, let's compare this modified equation to the second equation:
x + y = -3
5x + 5y = -15
We can see that the second equation is simply a multiple of the first equation (5 times the first equation). Thus, these equations represent the same line and are dependent.
This means that whenever we find a solution for the system, we can always find infinitely many solutions by finding any point on the line represented by the equations.
For example, let's solve the system of equations:
−x−y=3
5x+5y=−15
To solve the system, we can multiply the first equation by 5:
-5x - 5y = 15
5x + 5y = -15
Adding the two equations:
0 = 0
This equation is always true, which tells us that there are infinitely many solutions for this system. No matter what values we choose for x and y, the equations will always be true.
For instance, if we assign x = 1, then y can be any value, and the equations will still be true:
x = 1
−1−y=3
y = -4
So, one solution to the system is (1, -4). However, we can also choose any other value for y and still have a valid solution.
Overall, when a system of equations has infinitely many solutions, it means that the equations are dependent and represent the same line or family of parallel lines.
To determine if a system of equations has infinitely many solutions, we need to analyze the relationship between the given equations. In this case, let's examine the given system of equations:
Equation 1: -x - y = 3
Equation 2: 5x + 5y = -15
To check for infinitely many solutions, we will analyze whether the two equations are dependent or independent.
We can start by simplifying Equation 1 by multiplying both sides by -1:
x + y = -3
Now let's multiply Equation 1 by 5 to create an equivalent equation to Equation 2:
5(x + y) = 5(-3)
5x + 5y = -15
Notice that Equation 2 is equivalent to the simplified version of Equation 1. Since the two equations are the same, it means that they represent the same line graphically.
This situation indicates that the system has infinitely many solutions because the equations are dependent. Any values of x and y that satisfy one of the equations will also satisfy the other equation. In graphical terms, this means the lines formed by the equations are coincident or overlapping, resulting in infinite points of intersection.
To further demonstrate this concept, let's examine the equations and their graphs:
Equation 1: x + y = -3
Equation 2: 5x + 5y = -15
Equation 1 represents a straight line passing through the point (-3, 0) with a slope of -1. Equation 2, after simplification, gives the same line as Equation 1. Therefore, the two lines are coincident and overlap completely.
So, no matter what pair of values for x and y we choose that satisfy one of the equations, they will also satisfy the other equation. Consequently, the system has infinitely many solutions.
In conclusion, a system of equations has infinitely many solutions when the equations are dependent and represent coincident lines, leading to an infinite number of points of intersection.
When a system of equations has infinitely many solutions, it means that there are an infinite number of possible solutions that satisfy both equations. In other words, the equations are dependent and represent the same line on a graph.
To understand this concept, let's look at the given system of equations and see how they are related:
Equation 1: -x - y = 3
Equation 2: 5x + 5y = -15
To determine if the system has infinitely many solutions, we can use a few different methods. One way is to solve the equations and see what happens:
Step 1: Multiply Equation 1 by -5 to get rid of the coefficients of x in Equation 2:
5x + 5y = -15 (Equation 2)
5(-x - y) = 3(-5) (Equation 1 * -5)
which simplifies to:
-5x - 5y = -15
Step 2: Simplify Equation 2 by combining like terms:
-5x - 5y = -15
We can observe that both Equation 2 and the result of multiplying Equation 1 by -5 are the same. This indicates that the two equations represent the same line on a graph, meaning they have infinitely many solutions.
To further illustrate this, let's compare the equations in slope-intercept form (y = mx + b):
Equation 1: -x - y = 3 --> y = -x - 3
Equation 2: 5x + 5y = -15 --> y = -x - 3
As you can see, the two equations have the same slope (-1) and y-intercept (-3). This confirms that they represent the same line, and any values of x and y that satisfy one equation will also satisfy the other.
For example, let's choose an arbitrary value for x and solve for y in Equation 1 and Equation 2:
Let x = 1
Equation 1: y = -x - 3 --> y = -1 - 3 --> y = -4
Equation 2: y = -x - 3 --> y = -1 - 3 --> y = -4
As you can see, when x = 1, both equations have the same solution, y = -4. This demonstrates that there are infinitely many solutions to the system of equations.
In summary, a system of equations has infinitely many solutions when the equations are dependent and represent the same line. This can be determined by observing that the equations simplify to the same expression, have the same slope and y-intercept, or produce identical values when solved.