SOLUTION: solve each of the following systems by addition. if a unique solution does not exist, state whether the system is inconsistent or dependent. x + 5y = 10 and -2x -10y = -20

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: solve each of the following systems by addition. if a unique solution does not exist, state whether the system is inconsistent or dependent. x + 5y = 10 and -2x -10y = -20      Log On


   

Question 91420: solve each of the following systems by addition. if a unique solution does not exist, state whether the system is inconsistent or dependent.
x + 5y = 10 and -2x -10y = -20
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B5%2Ay=10
-2%2Ax-10%2Ay=-20

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and -2 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and -2 is -2, we need to multiply both sides of the top equation by -2 and multiply both sides of the bottom equation by -1 like this:

-2%2A%281%2Ax%2B5%2Ay%29=%2810%29%2A-2 Multiply the top equation (both sides) by -2
-1%2A%28-2%2Ax-10%2Ay%29=%28-20%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
-2%2Ax-10%2Ay=-20
2%2Ax%2B10%2Ay=20

Notice how -2 and 2 add to zero, -10 and 10 add to zero, -20 and 20 and to zero (ie -2%2B2=0) -10%2B10=0, and -20%2B20=0)


So we're left with

0=0


which means any x or y value will satisfy the system of equations. So there are an infinite number of solutions


So this system is dependent