Solving for x in Linear Equations

We are given a system of two linear equations with two variables, $x$ and $y$. Our goal is to find the specific value of $x$ that satisfies both equations simultaneously.

The given equations are:

• Equation 1: $3x + y = 12$
• Equation 2: $x - y = 4$

Method for Finding the Value of x

A straightforward method to solve this system and find the value of $x$ is the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.

In this case, we can see that the $y$ terms have opposite coefficients ($+1$ in Equation 1 and $-1$ in Equation 2). This makes elimination by addition very simple.

Step-by-Step Solution

1. Write down the equations:

   $3x + y = 12$ $x - y = 4$

2. Add the two equations:

   Align the equations vertically and add them term by term:

   $ \begin{array}{rcrcr} 3x & + & y & = & 12 \\ x & - & y & = & 4 \\ \hline \end{array} $

   Adding the corresponding terms:

   $ (3x + x) + (y - y) = (12 + 4) $

3. Simplify the resulting equation:

   The $y$ terms cancel each other out ($y - y = 0$), leaving us with an equation solely in terms of $x$.

   $ 4x + 0 = 16 $

   This simplifies to:

   $ 4x = 16 $

4. Solve for x:

   To isolate $x$, divide both sides of the equation $4x = 16$ by 4.

   $ x = \frac{16}{4} $

   Performing the division:

   $ x = 4 $

Conclusion

By using the elimination method, we have successfully found the value of $x$. The value of $x$ that satisfies both $3x + y = 12$ and $x - y = 4$ is 4.