How many solution(s) does the system of equations have?
−5x+y=−18
10x−2y=36(1 point)
This system of equations has one unique solution.
To see why, we can simplify the second equation by dividing both sides by 2:
10x - 2y = 36
becomes
5x - y = 18
Now we have two equations:
-5x + y = -18
5x - y = 18
Adding these equations eliminates the y terms:
0 = 0
This means the two equations are equivalent and represent the same line. In other words, the two equations have infinitely many solutions (every point on the line is a solution).
But if we look back at the original system, only one of the equations has a unique solution (the first equation). Therefore, the system as a whole has only one unique solution.
To determine the number of solutions for this system of equations, we can use the concept of linear independence.
Let's rewrite the system of equations in standard form:
Equation 1: -5x + y = -18
Equation 2: 10x - 2y = 36
First, let's simplify Equation 2 by dividing every term by 2:
0.5(10x - 2y) = 0.5(36)
5x - y = 18
Now, we can rewrite the system as follows:
Equation 1: -5x + y = -18
Equation 2: 5x - y = 18
Notice that the two equations are essentially equivalent; one equation is simply a multiple of the other. This implies that these equations represent the same line on a graph.
Since the equations represent the same line, they have an infinite number of solutions. Hence, the system has infinitely many solutions.
To determine the number of solutions for the given system of equations, we can solve the equations and see if they intersect at a single point, infinitely many points, or if they are parallel lines.
First, let's solve the system of equations:
Equation 1: -5x + y = -18
Equation 2: 10x - 2y = 36
We can use the method of substitution or elimination to solve the system. Let's use the elimination method:
Multiply Equation 1 by 2:
-10x + 2y = -36
Now, add Equation 2 and -10x + 2y = -36:
(10x - 2y) + (-10x + 2y) = 36 + (-36)
0 = 0
As we can see, the result is 0 = 0. This means that both equations are equivalent and represent the same line when graphed. Therefore, the system of equations has infinitely many solutions. The two equations are actually the same line, which means they intersect at every point along the line.
So, the answer is that the system of equations has infinitely many solutions.