Identify the solutions of the systems of equations, if any.
-4x-4y=8
4y=-4x-8
No solution
(0,-2)
(-2,0)
Infinitely many solutions
These lines are the same. There are an infinite numbers of solutions, a y for every x you pick.
To determine the solutions of a system of equations, we need to solve the equations simultaneously. Let's solve the given system of equations step by step.
Equation 1: -4x - 4y = 8
Equation 2: 4y = -4x - 8
First, let's rewrite Equation 2 by dividing both sides by 4 to isolate y:
4y/4 = (-4x - 8)/4
y = -x - 2
Now that we have the value of y in terms of x, we can substitute this expression into Equation 1 and solve for x:
-4x - 4(-x - 2) = 8
-4x + 4x + 8 = 8
8 = 8
Since 8 = 8 is a true statement, it means that the equation has infinitely many solutions. This means that any value of x will satisfy both equations.
To find the corresponding values of y, we can substitute the value of x back into either of the original equations. Let's use Equation 2:
4y = -4x - 8
4y = -4(0) - 8
4y = -8
y = -2
Therefore, the system of equations has infinitely many solutions, and the solution can be expressed as (x, y) = (any value of x, -2).
In conclusion, the solutions of the system of equations are infinitely many, and they can be represented as (x, y) = (any value of x, -2).
To identify the solutions of the system of equations, we can start by solving one equation for one variable and substituting it into the other equation. Let's solve the second equation for y:
4y = -4x - 8
By dividing both sides of the equation by 4, we get:
y = -x - 2
Now we can substitute this expression for y into the first equation:
-4x - 4(-x - 2) = 8
Simplifying this equation, we have:
-4x + 4x + 8 = 8
The variable x cancels out, leaving us with:
8 = 8
Since this equation is true, it means that the two equations represent the same line. Therefore, the system has infinitely many solutions, which means any point on the line is a solution.
So, the solution to the system is "infinitely many solutions."