Question

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

Answer 1

A system of equations can have infinitely many solutions when the equations are dependent on each other, meaning that one equation can be derived from another or they represent the same line.

In the given system of equations:
-x - y = 3
5x + 5y = -15

We can see that the second equation is simply the first equation multiplied by -5 on both sides. This means the second equation is dependent on the first equation, and both equations represent the same line.

To confirm this, we could rearrange the first equation to isolate either x or y. Let's isolate y:
-y = 3 + x
y = -3 - x

Now, if we substitute this expression for y into the second equation, we get:
5x + 5(-3 - x) = -15
5x -15 - 5x = -15
-15 = -15

As we can see, this equation is always true, regardless of the value of x. This means that any value of x and y that satisfies the first equation will also satisfy the second equation. Hence, the system of equations has infinitely many solutions.