I'm asked to solve the following system using elimination and say how many solutions there are:
4x - 7y = 15
-8x + 14y = -30
The answer says that there are infinitely many solutions, but I am getting 8x - 14y = 30 and -8x + 14y = -30. I think they both come up as 30 and -30, so they aren't the same. Could someone please explain how to do this?
Just multiply everything in either of your equations by -1 to see that they are really the same equation.
2x = 4
is the same as
-2x = -4
To solve this system of equations using elimination, you need to eliminate one variable by adding or subtracting the two equations. In this case, let's eliminate the x variable.
To do this, multiply the first equation by 2 to make the coefficients of x in both equations equal:
2(4x - 7y) = 2(15)
-8x + 14y = -30
Simplifying the equation gives us:
8x - 14y = 30 (equation 1)
-8x + 14y = -30 (equation 2)
Now, if we compare equation 1 and equation 2, we can observe that they are actually the same equation. The only difference is the signs on the right side. Therefore, both equations represent the same line, which means there are infinitely many solutions to the system.
In your attempt, you got -8x + 14y = -30 for the second equation, which is correct. However, you made an error in the signs when you tried to simplify the first equation. It should be 8x - 14y = 30, not 8x - 14y = -30.
Hence, the correct form of the system after elimination is:
8x - 14y = 30
-8x + 14y = -30
Since these equations are the same, they represent the same line, and therefore, there are infinitely many solutions to this system.