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We will solve the given system of equations using the elimination method. Before we can use this method, we will rewrite each of the equations in the system as a linear equation.
${2_{x+y}=164_{x−y}=1 (I)(II) $
Our goal is to write $16$ as a power of $2$ and to write $1$ as a power of $4.$ Then we will use the properties of exponents. Let's do it!
Next, we will rewrite $1$ as a power of $4.$ Recall that any non-zero number raised to the power of 0 is $1.$
In both equations we now have two equivalent expressions with the same base. If both sides of the equations are equal, the exponents must also be equal.
${2_{x+y}=2_{4}4_{x−y}=4_{0} ⇔{x+y=4x−y=0 $
Finally we will solve the obtained linear system of equations using the Elimination Method.
The solution to the given system of equations is $x=2,$ $y=2.$

${2_{x+y}=2_{4}4_{x−y}=1 $

$(II):$$1=a_{0}$

${2_{x+y}=2_{4}4_{x−y}=4_{0} $

${x+y=4x−y=0 $

Solve by elimination

${x+y+x−y=4+0x−y=0 $

AddSubTerms$(I):$Add and subtract terms

${2x=4x−y=0 $

${x=2x−y=0 $

${x=22−y=0 $

${x=22=y $

RearrangeEqn$(II):$Rearrange equation

${x=2y=2 $