Use the echelon method to solve the following system:
x/5+3y=31
2x-y/5=8
To solve the given system of equations using the echelon method, we need to eliminate a variable from one equation at a time until we have a system of equations that can easily be solved for each variable.
Let's start by getting rid of the fractions in the second equation. We can do this by multiplying both sides of the equation by 5 to clear the denominator:
5 * (2x - y/5) = 5 * 8
This simplifies to:
10x - y = 40
Now, we have the following system of equations:
x/5 + 3y = 31 (Equation 1)
10x - y = 40 (Equation 2)
To eliminate the variable "y" from the equations, we'll multiply Equation 1 by 5:
5 * (x/5 + 3y) = 5 * 31
This simplifies to:
x + 15y = 155
Now, we have the following system of equations:
x + 15y = 155 (Equation 3)
10x - y = 40 (Equation 2)
To eliminate the variable "x" from the equations, we'll multiply Equation 3 by 10:
10 * (x + 15y) = 10 * 155
This simplifies to:
10x + 150y = 1550
Now, we have the following system of equations:
10x + 150y = 1550 (Equation 4)
10x - y = 40 (Equation 2)
Next, we'll subtract Equation 2 from Equation 4:
(10x + 150y) - (10x - y) = 1550 - 40
This simplifies to:
10x + 150y - 10x + y = 1510
Combining like terms:
151y = 1510
Divide both sides of the equation by 151:
y = 1510 / 151
Simplifying this gives:
y = 10
Now that we have the value of "y", we can substitute it back into one of the original equations to find the value of "x". Let's use Equation 2:
10x - y = 40
Substituting y = 10, we get:
10x - 10 = 40
Adding 10 to both sides of the equation:
10x = 40 + 10
Simplifying:
10x = 50
Dividing both sides of the equation by 10:
x = 50 / 10
Simplifying:
x = 5
Therefore, the solution to the system of equations is x = 5 and y = 10.