How many solutions does the system of equations have? 8x−6y=48 8x+6y=48 (1 point) Responses none none two two one one infinite infinite
The system of equations is:
8x - 6y = 48
8x + 6y = 48
If we add the two equations together, the term with y will cancel out:
(8x - 6y) + (8x + 6y) = 48 + 48
16x = 96
x = 6
Substituting x = 6 into either of the original equations, we can solve for y:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
Therefore, the system of equations has one solution: (x, y) = (6, 0).
The correct answer is: one.
To determine the number of solutions, we can compare the slopes of the two lines formed by the given equations.
The given equations are:
1) 8x - 6y = 48
2) 8x + 6y = 48
To find the slope of each equation, let's rewrite them in slope-intercept form (y = mx + b):
1) 8x - 6y = 48
-6y = -8x + 48
y = (8/6)x - 8
y = (4/3)x - 8
2) 8x + 6y = 48
6y = -8x + 48
y = (-8/6)x + 8
y = (-4/3)x + 8
We see that both equations have the same slope, which is (4/3) and (-4/3) respectively. This means that the lines are parallel to each other.
When two parallel lines are considered, they do not intersect and have no common points. Therefore, the system of equations has no solution.
So, the correct response is: None.
To determine the number of solutions for a system of equations, we can compare the slopes of the two equations. If the slopes are equal and the y-intercepts are different, there are no solutions. If the slopes are equal and the y-intercepts are also equal, there are infinitely many solutions. If the slopes are different, there is a unique solution.
Let's compare the given system of equations:
Equation 1: 8x - 6y = 48
Equation 2: 8x + 6y = 48
Both equations have the same coefficients for x and y, which means the slopes are equal. Moreover, the y-intercepts are also equal since both equations are equal to 48.
Therefore, the system of equations has infinitely many solutions.