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.

Question

−x−y=3
5x+5y=−15

Answer 1

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.