how many solutions does 8x−6y=48 8x+6y=48 have?
i think one
You are correct. The two equations form a system of linear equations. By adding the two equations together, the y-variable cancels out:
(8x - 6y) + (8x + 6y) = 48 + 48
16x = 96
x = 96/16
x = 6
Substituting this value of x into either of the original equations, we find:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
Therefore, there is only one solution to the system of equations, which is x = 6 and y = 0.
To determine the number of solutions, let's analyze the system of equations:
1) 8x - 6y = 48
2) 8x + 6y = 48
We can use the method of addition or elimination to solve this system.
When we add the two equations together, we get:
(8x - 6y) + (8x + 6y) = 48 + 48
16x + 0y = 96
16x = 96
Dividing both sides of the equation by 16, we find that x = 6.
Now, substituting this value of x into either of the original equations, we can solve for y:
Using equation 1: 8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
Thus, the solution to the system of equations is x = 6 and y = 0.
Since there is only one unique solution, the system has one solution.
To determine the number of solutions for the system of equations:
1. Begin by rearranging the equations to isolate one variable (either x or y).
8x - 6y = 48 becomes: 8x = 6y + 48
8x + 6y = 48 remains as is
2. Now we have two equations:
Equation 1: 8x = 6y + 48
Equation 2: 8x + 6y = 48
3. Since the coefficient of x is the same in both equations (8), we can subtract Equation 1 from Equation 2 to eliminate x:
(8x + 6y) - (8x) = 48 - (6y + 48)
Simplifying, we get: 6y = -6y
4. This equation, 6y = -6y, tells us that 6y is equal to its negative value. In other words, the coefficient of y is effectively zero. This means that y can take on any value.
5. Since y can take on any value, we can substitute any value for y in Equation 1 to determine the corresponding value of x. This implies that there are infinitely many solutions.
Therefore, the system of equations 8x - 6y = 48 and 8x + 6y = 48 has infinitely many solutions.