The problem in your image involves finding the point where two lines intersect, with equations given as:

$y_1 = 2 - x \quad \text{and} \quad y_2 = 3x - 2$

Step 1: Set the equations equal to each other.

To find the intersection, we set $y_1 = y_2$ because at the point of intersection, both $y$-values are the same:

$2 - x = 3x - 2$

Step 2: Solve for $x$.

Rearrange the terms to solve for $x$:

$2 - x = 3x - 2 \implies 2 + 2 = 3x + x \implies 4 = 4x \implies x = 1$

Step 3: Substitute $x = 1$ into either equation to find $y$.

Substitute $x = 1$ into either $y_1 = 2 - x$ or $y_2 = 3x - 2$:

• Using $y_1 = 2 - x$:

$y_1 = 2 - 1 = 1$

Thus, the point of intersection is $(1, 1)$.

Step 4: Graph the lines.

Now, we will graph the two lines using their equations:

• For $y_1 = 2 - x$, this is a line with slope $-1$ and y-intercept $2$.
• For $y_2 = 3x - 2$, this is a line with slope $3$ and y-intercept $-2$.

Both lines will intersect at the point $(1, 1)$.

Would you like more details or help with graphing these lines?

Follow-up Questions:

1. Can you solve for the intersection of any two lines by setting the equations equal to each other?
2. What is the geometric significance of the slopes of the two lines in this problem?
3. How does changing the slope or intercept affect the intersection point?
4. Can you verify the solution graphically by plotting both lines on the grid?
5. How would you find the intersection if the lines were parallel or identical?

Tip: The intersection point of two lines represents the solution to the system of equations, as the point satisfies both equations simultaneously.