BThe Elimination Method can be used to solve a system of linear equations if either of the variable terms would cancel out the corresponding variable term in the other equation when added together.
ANo solution, see solution.
BInfinite solutions, see solution.
Practice makes perfect
To solve a system of linear equations using the Elimination Method, one of the variable terms needs to be eliminated when one equation is added to or subtracted from the other. This means that either the x- or the y-terms must cancel each other out.
x+4 y=2 & (I) x+4 y=10 & (II)
We can see that the x- and y-terms will eliminate each other if we subtract Equation (II) from Equation (I).
Solving this system of equations resulted in a contradiction, as 0 can never be equal to -8. This means that this system has no solutions.
We will solve the given system of equations using the Elimination Method. To do this, one of the variable terms needs to be eliminated when one equation is added to or subtracted from the other. This means that either the x- or the y-terms must cancel each other out.
2 x+4 y=-10 & (I) x+2 y=- 5 & (II)
Currently, none of the terms in this system will cancel out. Therefore, we need to find a common multiple between two variable like terms in the system. If we multiply Equation (II) by - 2, then the x- and y-terms will have opposite coefficients.
2 x+4 y=-10 - 2( x+2 y)=- 2(- 5)
⇓
2 x+4 y=-10 - 2 x-4 y=10
We can see that the x- and y-terms will eliminate each other if we add Equation (II) to Equation (I).
Solving this system of equations resulted in an identity — 0 is always equal to itself. Therefore, the system of equations has infinitely many solutions.
