To solve a using the , one of the terms needs to be eliminated when one is added to or subtracted from the other equation. This means that either the x-terms or the y-terms must cancel each other out.
2 x-3 y=-5 & (I) 5 x+2 y=16 & (II)
Currently, none of the terms in this system will cancel out. Therefore, we need to find a common multiple between two variable in the system. If we multiply Equation (I) by 2 and multiply Equation (II) by 3, the y-terms will have opposite .
2(2 x-3 y)=2(-5) & (I) 3(5 x+2 y)=3(16) & (II)
⇒
4 x-6 y=-10 15 x+6 y=48
We can see that the y-terms will eliminate each other if we add Equation (I) to Equation (II).
4x-6y=-10 15x+6y=48
4x-6y=-10 15x+6y+ 4x -6y=48+( -10)
4x-6y=-10 19x=38
4x-6y=-10 x=2
Now we can solve for y by substituting the value of x into either equation and simplifying. Let's use the first equation.
4x-6y=-10 x=2
4( 2)-6y=-10 x=2
8-6y=-10 x=2
-6y=-18 x=2
y=3 x=2
The solution, or , of the system of equations is (2,3).
Checking Our Solution
To check this solution, we will substitute it back into the given system and simplify. If doing so results in true statements for every equation in the system, our solution is correct.
2x-3y=-5 & (I) 5x+2y=16 & (II)
2( 2)-3( 3)? =-5 5( 2)+2( 3)? =16
4-9? =-5 10+6? =16
(I), (II): Add and subtract terms
-5=-5 16=16
Because both equations are true statements, we know our solution is correct.
