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The problem asks for the solution to the equation \( f(x) = g(x) \) where:

- \( f(x) = 2x + 7 \)
- \( g(x) = -x + 1 \)

To find the solution, we need to set \( f(x) = g(x) \) and solve for \( x \):

\[
2x + 7 = -x + 1
\]

Now, solve step by step:

1. Add \( x \) to both sides to move all \( x \)-terms to the left side:
   \[
   2x + x + 7 = 1
   \]
   This simplifies to:
   \[
   3x + 7 = 1
   \]

2. Subtract 7 from both sides:
   \[
   3x = 1 - 7
   \]
   \[
   3x = -6
   \]

3. Finally, divide both sides by 3:
   \[
   x = \frac{-6}{3}
   \]
   \[
   x = -2
   \]

Thus, the solution is \( x = -2 \).

Would you like more details or have any questions about the steps? Here are five related questions to explore:

1. How can you verify that \( x = -2 \) is the correct solution?
2. What is the graphical interpretation of the solution to \( f(x) = g(x) \)?
3. How would the solution change if the equation for \( g(x) \) was different?
4. What happens if you substitute \( x = -2 \) back into both functions?
5. How do you solve similar equations involving quadratic functions?

**Tip:** Always check your solution by substituting the value of \( x \) back into the original equations!

In order to solve the equation 2x + 7 = -x + 1, a student graphs two functions, f(x) = 2x + 7 and g(x) = -x + 1. What is the solution to the equation f(x) = g(x)?

Learn how to solve the linear equation 2x + 7 = -x + 1 by graphing two functions f(x) = 2x + 7 and g(x) = -x + 1. This problem involves key algebraic concepts such as graphing linear functions and solving systems of equations. Suitable for students in Grades 8-10.

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