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
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.
A system of equations having infinitely many solutions means that there are multiple sets of values that satisfy both equations simultaneously. In other words, the two equations describe the same line or are parallel.
Let's solve the given system of equations to see if it has infinitely many solutions.
To begin, let's rewrite the first equation:
-x - y = 3
=> y = -x - 3
Now we can substitute this expression for y into the second equation:
5x + 5(-x - 3) = -15
5x - 5x - 15 = -15
-15 = -15
As you can see, the result is a true statement (-15 = -15). This means that the second equation is redundant and doesn't provide any additional information. The two equations are essentially the same equation, just expressed differently.
Since both equations are describing the same line (or parallel lines), there are infinitely many sets of values that satisfy both equations. As an example, any point on the line y = -x - 3 would satisfy both equations.
Therefore, the given system of equations has infinitely many solutions.
When a system of equations has infinitely many solutions, it means that there are infinitely many pairs of values for the variables that satisfy all the equations in the system. In other words, the equations in the system can be satisfied simultaneously by an infinite number of values.
To determine whether a system of equations has infinitely many solutions, we can use various methods, such as graphing, substitution, or elimination. Let's use the elimination method for this particular system.
Given the system of equations:
1) -x - y = 3
2) 5x + 5y = -15
To solve this system using elimination, we can multiply equation 1) by 5 to make the coefficients of y in both equations equal:
5 * (-x - y) = 5 * 3
-5x - 5y = 15
Now, we can add equation 1) and the adjusted equation 5x + 5y = 15:
(-x - y) + (-5x - 5y) = 3 + 15
-6x - 6y = 18
The coefficients of x in both equations are equal, and the coefficients of y also match. This means that the two equations are equivalent, resulting in the same line when graphed. Therefore, the system of equations has infinitely many solutions.
To further illustrate this, let's rewrite the equations in terms of y:
1) -x - y = 3 => y = -x - 3
2) 5x + 5y = -15 => y = -x - 3
As we can see, both equations have the exact same expression for y. This means that for any value we substitute for x, we'll obtain the corresponding value of y that satisfies both equations. For example, if we choose x = 2, then y = -2 - 3 = -5. Alternatively, if we select x = -4, then y = -(-4) - 3 = 1. There are infinite combinations of x and y that satisfy both equations, hence infinitely many solutions.
Therefore, a system of equations having infinitely many solutions means that there are numerous pairs of values for the variables that satisfy all the equations in the system.