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=35x+5y=−15
A system of equations has infinitely many solutions when the two equations are dependent, meaning one equation is a multiple of the other. In this case, we can see that the second equation is equal to -5 times the first equation.
To confirm this, let's multiply the first equation by -5:
-5(-x - y) = -5(35)
5x + 5y = -175
We can see that the second equation is exactly the same as 5x + 5y = -175.
Since the two equations are equivalent, any values of x and y that satisfy one equation will automatically satisfy the other equation.
For example, let's choose x = 1. Plugging this into the first equation:
-1 - y = 3
y = -4
Now let's plug these values into the second equation:
1 + 5(-4) = -15
-19 = -15
The values x = 1 and y = -4 satisfy both equations. If we choose any other values for x and y that satisfy one equation, they will also satisfy the other equation. Therefore, the system of equations has infinitely many solutions.
To determine whether a system of equations has infinitely many solutions, we need to check for consistency and dependency among the equations. Let's analyze the given system of equations:
Equation 1: -x - y = 3
Equation 2: 5x + 5y = -15
We can first rearrange Equation 1 to isolate y:
-y = x + 3
y = -x - 3
Now let's substitute the value of y in Equation 2 with its equivalent expression from Equation 1:
5x + 5(-x - 3) = -15
5x - 5x - 15 = -15
-15 = -15
The equation simplifies to -15 = -15, which is true. This implies that both Equation 1 and Equation 2 are equivalent and represent the same line.
When two equations in a system are equivalent, it means that the equations have the same slope and intercept, resulting in infinite points of intersection or solutions. In other words, every point on the line of the equation is a solution to the system.
In this case, both Equation 1 and Equation 2 represent the same line and therefore have infinitely many solutions. Any (x, y) pair satisfying the equation -x - y = 3 or 5x + 5y = -15 will be a valid solution. For example, (0, -3), (1, -4), (-1, -2), etc., are all solutions to this system of equations.
When a system of equations has infinitely many solutions, it means that there are multiple sets of values for the variables that satisfy all the equations simultaneously. In other words, there are infinitely many combinations of values that can be substituted into the variables, and each combination will make both equations true.
To analyze the given system of equations {−x−y=35 and 5x+5y=−15, let's take a closer look at each equation:
Equation 1: −x−y=3 ... (1)
Equation 2: 5x+5y=−15 ... (2)
To find the solutions, we can solve the equations using the method of elimination or substitution:
Method of Elimination:
Multiply equation 1 by 5 to make the coefficients of 'y' the same in both equations:
-5x - 5y = 175 ... (3)
5x + 5y = -15 ... (2)
Now, add equation (3) and equation (2):
0 = 160
Since 0 is not equal to 160, we have reached an inconsistent result, indicating there are no solutions. However, this does not mean the system has only no solutions; it means it has infinitely many solutions.
To further illustrate this, let's plot the two equations on a graph:
Equation 1: −x−y=3
Rearranging the equation: y = -x - 3
By assigning different values to 'x', we can get the corresponding 'y' values. For example, when x = 0, y = -3, and when x = -3, y = 0. This creates a line on the graph.
Equation 2: 5x + 5y = -15
Rearranging the equation: y = -x - 3
This equation is the same as equation 1, which means both equations represent the same line. Thus, they coincide and overlap on the graph.
Since the two equations represent the same line, any point on that line will satisfy both equations. Therefore, there are infinitely many solutions to the system of equations.
In conclusion, a system of equations has infinitely many solutions when the equations represent the same line or planes in higher dimensions. This means there are multiple combinations of values for the variables that simultaneously satisfy all the equations.