Explain what it means for a system of equations to have infinitely many solutions. Use calculations and/or examples to justify your answer. Explain what it means for a system of equations to have infinitely mai examples to justify your answer.
-X - y = 3
5x + 5y = - 15
A system of equations has infinitely many solutions when the equations are dependent, meaning that one equation can be obtained from another by performing a series of valid operations (such as adding, subtracting, or multiplying by a constant). This typically occurs when the two equations represent the same line or planes in three-dimensional space.
To demonstrate this, let's analyze the given system of equations:
Equation 1: -x - y = 3
Equation 2: 5x + 5y = -15
We can find the corresponding values of x and y that satisfy both equations by solving them simultaneously. We'll use the method of substitution:
From Equation 1, we can rewrite y as follows:
y = -x - 3
Substituting this value of y into Equation 2:
5x + 5(-x - 3) = -15
5x - 5x - 15 = -15
-15 = -15
We obtain the equation -15 = -15, which is always true, regardless of the value of x. This indicates that the two equations in the system are actually the same line. Therefore, any value of x and y that satisfies one equation will also satisfy the other. The system has infinitely many solutions because every point on the line formed by these equations is a solution.
For example, let's assume x = 2. Substituting this value into Equation 1, we get:
-2 - y = 3
Solving for y:
y = -5
Thus, the ordered pair (2, -5) is a solution to the system of equations. However, we could choose any other value of x, and the corresponding y value that satisfies Equation 1 would also satisfy Equation 2. This illustrates that the system has infinitely many solutions, where every point on the line formed by these equations is a solution.