Question

An easy example of a linear problem

Answer 1

Sure! A classic example of a linear problem is finding the intersection of two lines in a two-dimensional space.

**Example: Finding the intersection of two lines**

Consider the following two linear equations:

1. \( y = 2x + 1 \)
2. \( y = -x + 4 \)

To find the intersection of these two lines, we need to solve for \( x \) and \( y \) where both equations are equal:

### Step 1: Set the equations equal to each other

Since both equations equal \( y \), we can set them equal to each other:

\[
2x + 1 = -x + 4
\]

### Step 2: Solve for \( x \)

Now, we can solve for \( x \):

1. Add \( x \) to both sides:
\[
2x + x + 1 = 4
\]
Simplifying gives:
\[
3x + 1 = 4
\]

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

3. Divide by 3:
\[
x = 1
\]

### Step 3: Substitute back to find \( y \)

Now, substitute \( x = 1 \) back into one of the original equations to find \( y \). Using the first equation:

\[
y = 2(1) + 1 = 2 + 1 = 3
\]

### Conclusion

The intersection of the two lines occurs at the point \( (1, 3) \).

This is a straightforward example of solving a linear problem involving two equations.