Solve each of the following systems, if possible. Indicate whether the system has a unique solution, infinitely many solutions, or no solution. I can either use substitution or elimination in this problem.
y=2-x
y=x-2
I did it already by using the substitution method and got an answer as (2,0). I just need to know if the system has a unique solution (I chose the unique solution option but I'm not sure if I'm right), infinitely many solutions, or no solution? I want to check with you guys if I'm doing this right! I started the problem like this x-2=2-x by substituting y=x-2. Please answer and don't ignore! :(
Since 2-x = -(x-2), you have y = -y
So, y=0
so 2-x = 0, and x=2
That is the only solution. You have two lines with different slopes, so the must intersect in exactly one point.
To determine the number of solutions for the given system of equations:
1. Start with the equations:
y = 2 - x ...(Equation 1)
y = x - 2 ...(Equation 2)
2. Since y is equal to both 2 - x and x - 2, we can set the two expressions equal to each other:
2 - x = x - 2
3. Simplify the equation:
Add x to both sides: 2 = 2x - 2
Add 2 to both sides: 4 = 2x
Divide both sides by 2: 2 = x
4. Now that we have x = 2, we can substitute this value back into either Equation 1 or Equation 2 to find the corresponding y-value.
Using Equation 1: y = 2 - 2 = 0.
5. The solution to the system of equations is (x, y) = (2, 0).
Based on the solution (2, 0), the system has a unique solution, meaning there is only one point of intersection for the two equations. Therefore, you are correct in choosing the unique solution option.
Well done on solving the system correctly!
To determine the solution to the system of equations, let's go through the steps using elimination method.
Given the system of equations:
1) y = 2 - x
2) y = x - 2
To solve this system using the elimination method, we need to eliminate one of the variables so that we can solve for the other variable. From the given equations, we can see that the coefficients of y in both equations are already the same (+1). So, let's eliminate y.
Subtract equation (2) from equation (1):
(1) - (2):
(y) - (y) = (2 - x) - (x - 2)
This simplifies to:
0 = 4
However, we have reached a contradiction, as 0 cannot equal 4. Therefore, the system of equations is inconsistent, meaning it has no solution.
Based on this, the system does not have a unique solution, nor does it have infinitely many solutions. It has no solution.
It seems like there might have been a mistake in the calculations using the substitution method when you obtained the point (2,0) as the solution. Double-check your calculations in the substitution method to ensure accuracy.