Use the method of elimination to solve the following system of equations. You must check your answer:
1)x/2 + y/5 = 1/2
2) 3x - y = -8
To solve the system of equations using the method of elimination, we need to eliminate one variable by adding or subtracting the two equations. Let's start by eliminating the y variable.
1) x/2 + y/5 = 1/2
2) 3x - y = -8
To eliminate the y variable, we can multiply equation 1 by 5 and equation 2 by 2:
5 * (x/2 + y/5) = 5 * (1/2)
2 * (3x - y) = 2 * (-8)
This gives us:
5x/2 + y = 5/2
6x - 2y = -16
Now, we can add the two equations together:
(5x/2 + y) + (6x - 2y) = (5/2) + (-16)
Simplifying, we get:
(5x/2 + 6x) + (y - 2y) = 5/2 - 16
Combining like terms, we have:
(5x/2 + 6x) - y = -11/2
Now, we can simplify further:
(11x/2) - y = -11/2
To eliminate the y variable, we can multiply equation 1 by 2:
2 * (11x/2 - y) = 2 * (-11/2)
This gives us:
11x - 2y = -11
Now, we can add the two equations together:
(11x/2 - y) + (11x - 2y) = (-11/2) + (-11)
Simplifying, we get:
(11x/2 + 11x) + (-y - 2y) = -11/2 - 11
Combining like terms, we have:
(11x/2 + 11x) - 3y = -33/2
Now, we have two equations:
(11x/2 - y) = -11/2
(11x/2 + 11x) - 3y = -33/2
We can divide the first equation by 2 to simplify it:
(11x/2)/2 - y/2 = (-11/2)/2
(11x/4) - y/2 = -11/4
Now, we can substitute this simplified equation into the second equation:
(11x/4) - y/2 + (11x/4) - 3y = -33/2
Combining like terms, we have:
(11x/2) - (11y/4) = -33/2
To eliminate the fractions, we can multiply both sides of the equation by 2:
2 * ((11x/2) - (11y/4)) = 2 * (-33/2)
This gives us:
11x - (11y/2) = -33
Now, we can divide both sides of the equation by 11 to isolate the x term:
(11x - (11y/2))/11 = (-33)/11
Simplifying, we get:
x - (y/2) = -3
To eliminate the y term, we can multiply equation 1 by 2:
2 * (x - (y/2)) = 2 * (-3)
This gives us:
2x - y = -6
Now, we have two equations:
2x - y = -6
11x - (11y/2) = -33
We can solve this system of equations by substitution or elimination.
Using substitution, we can solve equation 1 for x:
x = (-6 + y)/2
Substituting this value of x into equation 2:
11((-6 + y)/2) - (11y/2) = -33
Simplifying, we get:
-66 + 11y - 11y = -33
Since the y terms cancel out, we are left with:
-66 = -33
This equation is not true for any value of y. Therefore, there are no solutions to this system of equations.
We have checked our answer and determined that there are no solutions to the given system of equations.