Explain why the two equations below have the same solutions.
x + 3y = −1
−2x − 6y = 2
A. The two equations have the same slope, so they have the same solutions.
B. The second equation is a multiple of the first equation, so they have the same solutions.
C. The graphs of the equations are parallel and do not intersect, so any solution of one is a solution of the other.
D. The lines are perpendicular, so they have the same solutions.
Thank You very much I think a will be the most accurate answer is that correct or not. Please correct me if wrong I thank you for your time and patience!
let's look at the 2nd equation.
−2x − 6y = 2
multiply by -1
2x + 6y = -2
divide by 2
x + 3y = -1 , well, isn't that the first equation???
So, do you want to change your choice?
No, since it is right, right
No, your choice is not correct
It is B
If I multiply the first equation by -2, don't I get the 2nd equation???
So one is a multiple of the other.
Actually, (A) is not the correct answer. The statement "the two equations have the same slope" does not guarantee that they have the same solutions.
The correct answer is (B) - "the second equation is a multiple of the first equation, so they have the same solutions."
To understand why this is the case, we can use the concept of linear dependence. Two linear equations are said to be linearly dependent if one equation can be obtained by multiplying both sides of the other equation by a constant.
In this case, we can see that the second equation (-2x - 6y = 2) can be obtained by multiplying the first equation (x + 3y = -1) by -2. Therefore, the two equations are linearly dependent, and as a result, they have the same solutions.
To further demonstrate this, we can solve both equations to find their solutions. Let's start by solving the first equation.
x + 3y = -1
We can rearrange this equation to solve for x:
x = -1 - 3y
Now let's substitute this value of x into the second equation:
-2(-1 - 3y) - 6y = 2
Expanding the brackets:
2 + 6y - 6y = 2
Simplifying:
2 = 2
As we can see, the resulting equation is always true, no matter what value we assign to y. This indicates that any solution that satisfies the first equation (x + 3y = -1) will also satisfy the second equation (-2x - 6y = 2). Hence, the two equations have the same solutions.