How many solutions does the system of equations have?
8x - 6y = 48
8x + 6y = 48
A. infinite
B. one
C. none
D. two
The system of equations is as follows:
8x - 6y = 48
8x + 6y = 48
To find the number of solutions, we can check if the two equations are consistent and independent.
We can add the two equations together to eliminate the y-variable:
(8x - 6y) + (8x + 6y) = 48 + 48
16x = 96
x = 6
Now, we can substitute the value of x back into either of the original equations to solve for y:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
The system has only one solution, x = 6 and y = 0.
Therefore, the answer is B. one.
To find the number of solutions the system of equations has, we can analyze the coefficients of x and y.
In this case, the coefficients of y are -6 in the first equation and 6 in the second equation. Since the coefficients of y have opposite signs, the system of equations represents two parallel lines.
Since these lines are parallel, they will never intersect, meaning there are no common solutions. Therefore, the system of equations has (C) none solutions.
To determine the number of solutions for this system of equations, we can compare the slopes of the two equations.
First, let's rearrange the equations to put them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
Equation 1: 8x - 6y = 48
Rearranging, we get: -6y = -8x + 48
Dividing everything by -6, we get: y = (4/3)x - 8
Equation 2: 8x + 6y = 48
Rearranging, we get: 6y = -8x + 48
Dividing everything by 6, we get: y = (-4/3)x + 8
Now, let's compare the slopes of the two equations. The slopes are the coefficients of x in each equation.
For Equation 1: y = (4/3)x - 8, the slope is 4/3.
For Equation 2: y = (-4/3)x + 8, the slope is -4/3.
Since the slopes are not equal, this means the two lines are not parallel and therefore intersect at a single point.
Therefore, the system of equations has only one solution.
The answer is B. one.