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

1 answer

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.