To determine how many solutions this system of equations has, we will solve it using the Elimination Method. Doing so will result in one of three cases.
This means we should solve the given system of equations, then make our conclusion based on the result.
Solve by Elimination
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 equation. In this exercise, this means that either the x-terms or the y-terms must cancel each other out.
4x+ y=21 & (I) -2 x+ 6y=9 & (II)
In its current state, this will not happen. Therefore, we need to find a common multiple between two variable like terms in the system. If we multiply Equation (II) by 2 the x-terms will have opposite coefficients.
4 x+ y=21 2(-2 x+6 y)=2(9) ⇒ 4x+ y=21 -4 x+12 y=18
With this, we can see that the x-terms will eliminate each other if we add Equation (I) to Equation (II).
