Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
1/5x -1/4 y = 3
2/5x + 1/2y = 2
first one times 20
4x - 5y = 60
second one times 10
4x + 5y = 20
add them
8x = 80
x = 10
in 4x + 5y = 20
if x = 10, y = -4
To solve this system of equations using the elimination method, we need to eliminate one of the variables by adding or subtracting the equations. Let's start by eliminating the y variable.
To do this, we'll multiply the first equation by 2 and the second equation by -1 to make the coefficients of y in both equations equal:
Equation 1: (2) * (1/5x) - (2) * (1/4y) = (2) * (3)
Equation 2: (-1) * (2/5x) + (-1) * (1/2y) = (-1) * (2)
This simplifies the two equations to:
2/5x - 1/2y = 6
-2/5x - 1/2y = -2
Now, let's add the two equations together to eliminate the y variable:
(2/5x - 1/2y) + (-2/5x - 1/2y) = 6 + (-2)
When we add the equations, the x terms cancel out, leaving us with:
0 - 1y = 4
Simplifying further, we get:
-1y = 4
To solve for y, we divide both sides of the equation by -1:
y = -4
Now, substitute the value of y back into one of the original equations. Let's use the first equation:
1/5x - 1/4(-4) = 3
Simplifying and solving for x:
1/5x + 1 = 3
1/5x = 3 - 1
1/5x = 2
x = 2 * 5
x = 10
Therefore, the solution to the system of equations is x = 10 and y = -4.
There is only one solution to this system.