do the Two systems of equations shown have the same solution?
4x+y=−1
x+y=2
8x+2y=−2
−4x+2y=−8
Yes, the two systems of equations have the same solution. The first system of equations can be solved by adding the two equations, resulting in:
4x + y + x + y = -1 + 2
5x + 2y = 1
Solving the second system of equations also by adding the two equations will result in the same equation:
8x + 2y + (-4x) + 2y = -2 + (-8)
4x + 4y = -10
So, both systems of equations lead to the equation 5x + 2y = 1, indicating that they have the same solution.
To determine if the two systems of equations have the same solution, we need to solve each system and compare the solutions.
Let's start with the first system:
1) 4x + y = -1
2) x + y = 2
We can solve this system using the method of substitution or elimination. Let's use the method of elimination by adding the equations together:
1) 4x + y = -1
2) x + y = 2
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3) 5x + 2y = 1
Now, let's solve the second system:
1) 8x + 2y = -2
2) -4x + 2y = -8
Again, let's use the method of elimination by adding the equations together:
1) 8x + 2y = -2
2) -4x + 2y = -8
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3) 4x + 4y = -10
By comparing the third equations of both systems:
- For the first system: 5x + 2y = 1
- For the second system: 4x + 4y = -10
We can see that the third equations in both systems are different, which means the two systems have different solutions. Therefore, the two systems of equations shown do not have the same solution.
To determine if the two systems of equations have the same solution, we can solve them separately and compare the solutions.
Let's solve the first system of equations:
4x + y = -1 ...(Equation 1)
x + y = 2 ...(Equation 2)
We can solve this system by elimination or substitution method. I will use the substitution method.
From Equation 2, we have x = 2 - y.
Substituting this value of x into Equation 1:
4(2 - y) + y = -1
8 - 4y + y = -1
-3y = -9
y = 3
Substituting the value of y back into equation 2:
x + 3 = 2
x = -1
Therefore, the solution for the first system of equations is x = -1, y = 3.
Now, let's solve the second system of equations:
8x + 2y = -2 ...(Equation 3)
-4x + 2y = -8 ...(Equation 4)
Again, we can use the substitution method. From Equation 4, we have -4x = -8 - 2y, which simplifies to -2x = -4 - y or y = -2x - 4.
Substituting this value of y into Equation 3:
8x + 2(-2x - 4) = -2
8x - 4x - 8 = -2
4x - 8 = -2
4x = 6
x = 6/4
x = 3/2
Substituting the value of x back into Equation 4:
-4(3/2) + 2y = -8
-6 + 2y = -8
2y = -2
y = -1
Therefore, the solution for the second system of equations is x = 3/2, y = -1.
Since the solutions for the two systems of equations are different, the two systems do not have the same solution.