find all solutions of the given systems of equations   7x + y = 6
−7x + y = 6

To solve this system of linear equations:

\[
7x + y = 6
\]
\[
-7x + y = 6
\]

we can use the elimination method or substitution method. Here, we'll use elimination.

1. **Rewrite the system of equations:**

   \[
   7x + y = 6 \quad \text{(Equation 1)}
   \]
   \[
   -7x + y = 6 \quad \text{(Equation 2)}
   \]

2. **Eliminate \( x \):**  
   Add Equation 1 and Equation 2:

   \[
   (7x + y) + (-7x + y) = 6 + 6
   \]

   Simplifying this:

   \[
   0x + 2y = 12
   \]

   which reduces to:

   \[
   2y = 12
   \]

3. **Solve for \( y \):**

   Divide by 2:

   \[
   y = 6
   \]

4. **Substitute \( y = 6 \) back into one of the original equations to find \( x \):**

   Using Equation 1:

   \[
   7x + 6 = 6
   \]

   Subtract 6 from both sides:

   \[
   7x = 0
   \]

   Divide by 7:

   \[
   x = 0
   \]

5. **Solution:**

   The solution to the system is:

   \[
   (x, y) = (0, 6)
   \]

---

Would you like further explanation on any step?

1. What is elimination in solving systems of equations?
2. How do we know when to use substitution vs. elimination?
3. Why does adding the equations help in elimination?
4. How can we confirm our solution for a system of equations?
5. How would we solve if the system were nonlinear?

**Tip:** When two equations look similar but have opposite signs for one variable, elimination can often simplify the solution process quickly.

Find all solutions of the given system of equations: 7x + y = 6 and −7x + y = 6.

Learn how to solve the system of linear equations 7x + y = 6 and −7x + y = 6 step-by-step using the elimination method. This solution covers key algebra concepts, including elimination, linear equations, and substitution techniques. Suitable for students in Grades 8-10.

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