A line 1 passes through the point (3,3) and (5,7). Another line 2 passes through (-2,3). Find the x-intercept of line 2 and determine the point of intersection of line 1 and 2.

Let's solve the problem step by step.

### Step 1: Find the equation of Line 1
Line 1 passes through the points \( (3, 3) \) and \( (5, 7) \).

1. **Find the slope** of Line 1:
   The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the given points \( (3, 3) \) and \( (5, 7) \):
   \[
   m = \frac{7 - 3}{5 - 3} = \frac{4}{2} = 2
   \]

2. **Use the point-slope form** to find the equation of Line 1:
   The point-slope form of the equation of a line is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Using the point \( (3, 3) \) and the slope \( m = 2 \):
   \[
   y - 3 = 2(x - 3)
   \]
   Simplify to get the equation in slope-intercept form:
   \[
   y - 3 = 2x - 6
   \]
   \[
   y = 2x - 3
   \]
   So, the equation of Line 1 is:
   \[
   y = 2x - 3
   \]

### Step 2: Find the equation of Line 2
Line 2 passes through the point \( (-2, 3) \). We need to find the x-intercept of this line, which means we need the full equation of the line.

1. **Assume the slope of Line 2 is \( m \)** and use the point-slope form:
   \[
   y - y_1 = m(x - x_1)
   \]
   Using the point \( (-2, 3) \):
   \[
   y - 3 = m(x + 2)
   \]
   To find the x-intercept, set \( y = 0 \) and solve for \( x \):
   \[
   0 - 3 = m(x + 2)
   \]
   \[
   -3 = m(x + 2)
   \]
   \[
   x = -\frac{3}{m} - 2
   \]
   The x-intercept of Line 2 depends on the value of \( m \). To proceed further, we need to either know the slope or solve it using another approach (e.g., parallel or perpendicular criteria with other lines). Since only one point on Line 2 is provided, we'll make assumptions based on given details later.

### Step 3: Find the Point of Intersection of Line 1 and Line 2
Assume that the slope \( m \) is such that \( Line 2 \) does not end up parallel to \( Line 1 \). The equation of Line 2 intersects Line 1 at some point, denoted by \( P(x, y) \). Set up the system of linear equations and solve:

1. **Assume** \( m \) to calculate the y-intercept:
   \[
   y = mx + b
   \]
   Since \( y = 3 \) at \( x = -2 \), we solve for \( b \):
   \[
   3 = m(-2) + b
   \]
   \[
   b = 3 + 2m
   \]
   Substituting \( b \) into the general equation:
   \[
   y = mx + 2m + 3
   \]

2. **Equating Line 1 and Line 2** to find the point of intersection:
   The line equations would simplify to:
   \[
   2x - 3 = mx + 2m + 3
   \]
   Solving for \( x \) when equating these two results would provide the needed coordinates for the point of intersection.

I realize this explanation may have reached a conceptual block; let's focus back and assume a more straightforward calculation approach that doesn't assume on specific numbers. For ease, Line 2 is often presented as a vertical/horizontal line, so assume certain symmetrical properties. Would you like to continue with this approach or something simpler?

### Step 4: Finalize both intercept and point

- **X-intercept of Line 2**: Use final simplified intercept form to resolve.
- **Intersection Point**: Reiterate with the earlier y-intercept if no confusion.

Would you like to refine or re-approach solving? Happy to detail steps!

1. How would the solution change if we knew the slope of Line 2?
2. Could Line 1 or Line 2 be parallel based on these assumptions?
3. What if they were perpendicular?
4. How to represent the problem on a graph?
5. Could substitution or elimination better simplify calculations?

**Tip:** Always review the problem's intended format of results before solving to avoid confusions!

Learn how to solve a problem involving two intersecting lines. Find the x-intercept of one line and determine the point of intersection between two lines using step-by-step calculations. Suitable for Grades 7-9 students.

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