Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
-8x – 2y = -28
-3x + 5y = 47
5(-8x-2y)=-28 * 5
2(-3x * 5y)=47 * 2
After solved y should be cancelled out.
To solve the given system of equations using the elimination method, we need to eliminate one variable by manipulating the equations so that when added or subtracted, one variable cancels out. Let's start by manipulating the equations to eliminate the x variable.
First, we'll multiply the second equation by 8 to make the coefficients of the x variable the same in both equations:
-8x – 2y = -28
-3x + 5y = 47
(8)(-3x + 5y) = 8(47)
-24x + 40y = 376
Now, we can add the two equations together to eliminate the x variable:
-8x – 2y + (-24x + 40y) = -28 + 376
-8x - 24x - 2y + 40y = 348
-32x + 38y = 348
Now, let's continue to eliminate a variable by manipulating the equations to eliminate the y variable. We'll multiply the first equation by 38:
(38)(-8x – 2y) = 38(-28)
-304x - 76y = -1064
Now, we can add the newly multiplied first equation to the second equation:
-304x - 76y + (-32x + 38y) = -1064 + 348
-304x - 32x - 76y + 38y = -716
-336x - 38y = -716
Now, we have a new equation: -336x - 38y = -716. This equation represents the same line as the previous one: -32x + 38y = 348. When we subtract these two equations, we get:
(-336x - 38y) - (-32x + 38y) = -716 - 348
-336x - 38y + 32x - 38y = -1064
-304x - 76y = -1064
As you can see, we end up with the same equation as before. This means that the two original equations are dependent, and there's an infinite number of solutions.
In summary, the given system of equations has an infinite number of solutions.