Question

How do you solve for $y$:$2x + y = 8,{\text{ }}2x - y = 8$?

Answer 1

Hint: Make the coefficient of any one of the variables the same by multiplying the equations with respective constants.
Add or subtract them to eliminate that variable from the equation.
Find the value of one variable from the new equation.
Substitute the value of this variable in one of the two original equations to get the second variable

Complete step-by-step solution:
Notice that the $y$ terms are additive inverses. Add the two equations together to eliminate them.
The given equations are $2x + y = 8,2x - y = 8$
$2x + y = 8 \to (1)$
$2x - y = 8 \to (2)$
Now we add the two equations $(1){\text{ and (2)}}$, hence we get
$2x + y =8$ + $2x - y = 8$
$\Rightarrow$$2x + y + 2x - y = 8 + 8$
$\Rightarrow$$4x + 0 = 16$
The zero terms vanish, hence we get
$\Rightarrow$$4x = 16$
Divide by $4$ on both sides, hence we get
$\Rightarrow$\[\dfrac{{\not{4}}}{{\not{4}}}x = \dfrac{{16}}{4}\]
Divide $16$ by $4$, hence we get
$\Rightarrow$$x = 4$
The $x$ value substitute in the first equation $(1)$, hence we get
$\Rightarrow$$2x + y = 8$
If $x = 4$, then
$\Rightarrow$$2(4) + y = 8$
Multiply $2$ by $4$, hence we get
$\Rightarrow$$8 + y = 8$
Now, we subtract by $8$ on both sides, hence we get
$\Rightarrow$ $8 - 8 + y = 8 - 8$
On simplified
$\Rightarrow$$0 + y = 0$
The zero terms vanish, hence we get
$\Rightarrow$$y = 0$

Therefore the value of y is equal to 0.

Note: We could have guessed that from looking at the equation. In one $y$ is added and in the other $y$ is subtracted, yet it does not affect the value of the equation at all. The only value of $y$that would allow this to happen would be $y = 0$. If you end with $0 = 0$, then it means that the left-hand side and the right-hand side of the equation are equal to each other regardless of the values of the variables involved; therefore, its solution set is all real numbers for each variable.