How many solutions does the system of equations have?
8x−6y=48
8x+6y=48 (1 point)
Responses
none
none
infinite
infinite
two
two
one
one
bot gpt and step by step bot
To find the number of solutions in the given system of equations:
8x - 6y = 48 ...(1)
8x + 6y = 48 ...(2)
We can observe that equation (1) and equation (2) have the same coefficients and constant term except for the sign of the y-term. This means that the two equations represent the same line.
Since the two equations represent the same line, they have an infinite number of solutions. This implies that the correct answer is "infinite."
To determine the number of solutions for the given system of equations:
1. Write down the equations:
Equation 1: 8x - 6y = 48
Equation 2: 8x + 6y = 48
2. Notice that the coefficients in front of y in both equations are the same: -6 in Equation 1 and 6 in Equation 2.
3. Add the two equations together to eliminate the y-terms:
(8x - 6y) + (8x + 6y) = 48 + 48
16x = 96
4. Divide both sides of the equation by 16 to solve for x:
x = 96/16
x = 6
5. Substitute the value of x into one of the original equations to solve for y. Let's use Equation 1:
8(6) - 6y = 48
48 - 6y = 48
-6y = 0
y = 0
6. The solution to the system of equations is x = 6 and y = 0.
Therefore, the system of equations has one solution.
To determine the number of solutions for a system of equations, we can compare the slopes of the two equations.
The given system of equations is:
1) 8x - 6y = 48
2) 8x + 6y = 48
To compare the slopes, let's rewrite the equations in slope-intercept form (y = mx + b), where m represents the slope:
1) 8x - 6y = 48
-6y = -8x + 48
y = (8/6)x - 8
2) 8x + 6y = 48
6y = -8x + 48
y = (-8/6)x + 8
Now we can compare the slopes of the two equations.
In equation 1, the slope is 8/6, which can be simplified to 4/3.
In equation 2, the slope is -8/6, which can be simplified to -4/3.
Since the slopes are different, the lines represented by these equations are neither parallel nor coincident. Therefore, they intersect at a single point and have one solution.
The answer to the question "How many solutions does the system of equations have?" is one solution.