How many solution(s) does the system of equations have? −5x+y=−18 10x−2y=36 (1 point) Responses infinite infinite none none two two one
The system of equations is inconsistent and does not have a solution. The answer is none.
To determine the number of solutions for a system of equations, we can utilize the concept of slope-intercept form (y = mx + b) and compare the slopes and y-intercepts.
Given the system of equations:
-5x + y = -18 ...(Equation 1)
10x - 2y = 36 ...(Equation 2)
First, let's rewrite Equation 1 in slope-intercept form:
y = 5x - 18
The slope (m1) of this equation is 5.
Now let's rewrite Equation 2 in slope-intercept form:
2y = 10x - 36
y = 5x - 18
The slope (m2) of this equation is also 5.
Since the slopes (m1 and m2) are the same, the lines are parallel, and there are no intersection points. This means the system of equations has no solution.
Therefore, the correct response is: none
To determine the number of solutions for a system of equations, you need to solve the equations and see how many distinct solutions are obtained.
Let's solve the given system of equations:
Equation 1: −5x + y = −18
Equation 2: 10x − 2y = 36
We can solve this system of equations using the method of elimination or substitution. Let's use the method of elimination:
To use the elimination method, we need to multiply both sides of Equation 1 by 2 to make the coefficients of 'y' in both equations equal.
Multiply Equation 1 by 2:
2(-5x) + 2y = 2(-18)
-10x + 2y = -36
Now, we have the following system of equations:
-10x + 2y = -36
10x - 2y = 36
When we add the two equations together, the variable 'y' cancels out:
(-10x + 2y) + (10x - 2y) = -36 + 36
0 = 0
We obtain an equation '0 = 0', which means that both equations are equivalent and represent the same line. This indicates that the two equations are dependent, and therefore, they have infinite solutions.
Hence, the system of equations has an infinite number of solutions.