How many solutions does the system of equations have 8x-6y=48 and 8x+6y=48
To solve the system of equations, we can add the two equations together to eliminate the y terms:
(8x - 6y) + (8x + 6y) = 48 + 48
16x = 96
x = 6
Substituting x = 6 into one of the equations, we can solve for y:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
Therefore, the system of equations has one unique solution, (x, y) = (6, 0).
To determine the number of solutions for this system of equations, we need to compare the slopes of the two lines.
Let's start by rearranging the equations in slope-intercept form (y = mx + b), where m is the slope:
Equation 1: 8x - 6y = 48
Rearranging, we get: -6y = -8x + 48
Dividing through by -6, we have: y = (8/6)x - 8
Equation 2: 8x + 6y = 48
Rearranging, we get: 6y = -8x + 48
Dividing through by 6, we have: y = (-8/6)x + 8
Now we can compare the slopes:
The slope of Equation 1 is (8/6) = 4/3.
The slope of Equation 2 is (-8/6) = -4/3.
Since the slopes are equal in magnitude but opposite in sign, the lines are parallel. Parallel lines do not intersect and have no solutions in common.
Therefore, the system of equations has no solutions.
To determine the number of solutions a system of equations has, you need to compare the coefficients of the variables and constant terms.
Let's start by writing down the given system of equations:
1) 8x - 6y = 48
2) 8x + 6y = 48
Notice that the coefficients of y in each equation are opposites (-6y and +6y). This means that the two equations represent two parallel lines on a graph. Parallel lines will never intersect, which implies that the system of equations has no solution, or is inconsistent.
You can also verify this mathematically by adding the two equations together to eliminate the y variable.
Add equation 1 and equation 2:
(8x - 6y) + (8x + 6y) = 48 + 48
8x + 8x = 96
16x = 96
x = 6
Now, substitute the value of x (6) back into either of the original equations:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
Since both x and y equal to a specific value (x = 6, y = 0), the system would have a unique solution. However, this solution does not satisfy both equations simultaneously, indicating that the system of equations is inconsistent and has no solutions.