To solve the problem step by step, we will follow the instructions given in the question.
### Step 1: Convert the equations to slope-intercept form
We start with the two equations:
1. \(3x - y - 2 = 0\)
2. \(2x + y - 8 = 0\)
We will rearrange these equations to express \(y\) in terms of \(x\).
For the first equation:
\[
3x - y - 2 = 0 \implies y = 3x - 2
\]
For the second equation:
\[
2x + y - 8 = 0 \implies y = 8 - 2x
\]
### Step 2: Choose values for \(x\) and calculate corresponding \(y\) values
We will select three values for \(x\) for each equation and calculate the corresponding \(y\) values.
**For the first equation \(y = 3x - 2\)**:
- If \(x = 0\), then \(y = 3(0) - 2 = -2\) → Point (0, -2)
- If \(x = 1\), then \(y = 3(1) - 2 = 1\) → Point (1, 1)
- If \(x = 2\), then \(y = 3(2) - 2 = 4\) → Point (2, 4)
**For the second equation \(y = 8 - 2x\)**:
- If \(x = 0\), then \(y = 8 - 2(0) = 8\) → Point (0, 8)
- If \(x = 1\), then \(y = 8 - 2(1) = 6\) → Point (1, 6)
- If \(x = 2\), then \(y = 8 - 2(2) = 4\) → Point (2, 4)
- If \(x = 3\), then \(y = 8 - 2(3) = 2\) → Point (3, 2)
### Step 3: Plot the points on graph paper
Using graph paper, plot the points calculated for both equations:
- For \(y = 3x - 2\): (0, -2), (1, 1), (2, 4)
- For \(y = 8 - 2x\): (0, 8), (1, 6), (2, 4), (3, 2)
### Step 4: Draw the lines
Connect the points for each equation to form straight lines. The first line will connect the points (0, -2), (1, 1), and (2, 4). The second line will connect the points (0, 8), (1, 6), (2, 4), and (3, 2).
### Step 5: Find the point of intersection
From the graph, observe where the two lines intersect. The intersection point can be determined from the plotted points:
- The lines intersect at the point (2, 4).
### Step 6: Calculate the area of the triangle formed by the lines and the x-axis
The triangle is formed by the x-axis and the points of intersection with the lines. The vertices of the triangle are:
- (0, 0) (the origin)
- (2, 0) (the x-intercept of the line \(y = 3x - 2\))
- (2, 4) (the intersection point)
To find the area of the triangle, we can use the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is the distance along the x-axis from (0, 0) to (2, 0), which is 2 units. The height is the y-coordinate of the intersection point (2, 4), which is 4 units.
Thus, the area is:
\[
\text{Area} = \frac{1}{2} \times 2 \times 4 = 4 \text{ square units}
\]
### Final Answers
1. The coordinates of the point of intersection are \((2, 4)\).
2. The area of the triangle formed by the lines and the x-axis is \(4\) square units.
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