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Task 1: Draw the lines

First, let's draw the lines on the coordinate plane:

1. $x = 8$: This is a vertical line that passes through the point $(8, 0)$ on the x-axis.
2. $y = 3$: This is a horizontal line that passes through the point $(0, 3)$ on the y-axis.
3. $y = -x$: This is a straight line that passes through the origin $(0, 0)$ and has a slope of $-1$. This means for every 1 unit you move to the right on the x-axis, you move 1 unit down on the y-axis.

Task 2: Calculate the area of the triangle

The lines $x = 8$, $y = 3$, and $y = -x$ form a triangle. To find the area of this triangle, we need to determine the coordinates of its vertices.

1. Intersection of $x = 8$ and $y = 3$: The point of intersection is $(8, 3)$.
2. Intersection of $x = 8$ and $y = -x$: Substituting $x = 8$ into $y = -x$, we get $y = -8$. So the point of intersection is $(8, -8)$.
3. Intersection of $y = 3$ and $y = -x$: Substituting $y = 3$ into $y = -x$, we get $3 = -x$, so $x = -3$. The point of intersection is $(-3, 3)$.

Now we have the vertices of the triangle: $(8, 3)$, $(8, -8)$, and $(-3, 3)$.

We can use the formula for the area of a triangle given the coordinates of its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$:

$Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

Plugging in the coordinates:
$Area = \frac{1}{2} |8(-8 - 3) + 8(3 - 3) + (-3)(3 - (-8))|$
$Area = \frac{1}{2} |8(-11) + 8(0) - 3(11)|$
$Area = \frac{1}{2} |-88 + 0 - 33|$
$Area = \frac{1}{2} |-121|$
$Area = \frac{1}{2} \cdot 121$
$Area = 60.5$

So, the area of the triangle is 60.5 square units.

Final Answer: The area of the triangle enclosed by the lines is 60.5 square units.

Okay, let's proceed with the next task.

Task 3: Find the fourth point to form a square

We are given three points on the coordinate grid and need to find a fourth point that forms a square. Let's denote the given points as follows:
* $A = (1, 4)$
* $B = (1, 1)$
* $C = (4, 1)$

To form a square, the sides must be equal in length and the angles must be 90 degrees. We can analyze the given points to determine the possible locations of the fourth point, $D$.

1. Side AB: The length of side $AB$ is $4 - 1 = 3$ units (vertical).
2. Side BC: The length of side $BC$ is $4 - 1 = 3$ units (horizontal).

Since $AB$ and $BC$ are perpendicular and equal in length, we can assume that $A$, $B$, and $C$ are consecutive vertices of the square. Therefore, we need to find point $D$ such that $AD$ is parallel to $BC$ and $CD$ is parallel to $AB$.

To find the coordinates of point $D$, we can move from point $A$ horizontally by the same distance as $BC$, but in the opposite direction. Since $C$ is to the right of $B$, $D$ will be to the left of $A$. The x-coordinate of $A$ is 1, so we subtract 3 to get $1 - 3 = -2$. The y-coordinate of $D$ will be the same as $A$, which is 4.

Therefore, the coordinates of point $D$ are $(-2, 4)$.

Final Answer: The coordinates of the fourth point are $(-2, 4)$.

Okay, let's proceed with the next task.

Task 4: Create a table of $x$ and $y$ values for the line $y = 4x + 2$

We need to create a table of $x$ and $y$ values for the line $y = 4x + 2$, where $x$ ranges from $-1$ to $3$.

Here's how we can calculate the $y$ values for each $x$:

• For $x = -1$:
  $y = 4(-1) + 2 = -4 + 2 = -2$
• For $x = 0$:
  $y = 4(0) + 2 = 0 + 2 = 2$
• For $x = 1$:
  $y = 4(1) + 2 = 4 + 2 = 6$
• For $x = 2$:
  $y = 4(2) + 2 = 8 + 2 = 10$
• For $x = 3$:
  $y = 4(3) + 2 = 12 + 2 = 14$

Now, let's create the table:

Final Answer: The table of $x$ and $y$ values for the line $y = 4x + 2$ is shown above.

Okay, let's proceed with the next task.

Task 5: Create a table of $x$ and $y$ values for the line $y = 8x + 4$

We need to create a table of $x$ and $y$ values for the line $y = 8x + 4$, where $x$ ranges from $-1$ to $3$.

Here's how we can calculate the $y$ values for each $x$:

• For $x = -1$:
  $y = 8(-1) + 4 = -8 + 4 = -4$
• For $x = 0$:
  $y = 8(0) + 4 = 0 + 4 = 4$
• For $x = 1$:
  $y = 8(1) + 4 = 8 + 4 = 12$
• For $x = 2$:
  $y = 8(2) + 4 = 16 + 4 = 20$
• For $x = 3$:
  $y = 8(3) + 4 = 24 + 4 = 28$

Now, let's create the table:

Final Answer: The table of $x$ and $y$ values for the line $y = 8x + 4$ is shown above.

Okay, I will solve the problems step by step.

Task 6: Complete the table of values for $y = -x + 1$

We need to complete the table of values for the line $y = -x + 1$. Let's find the values for $A$ and $B$.

• For $x = -2$ (Value A):
  $y = -(-2) + 1 = 2 + 1 = 3$
  So, $A = 3$
• For $x = 1$ (Value B):
  $y = -(1) + 1 = -1 + 1 = 0$
  So, $B = 0$

Now, let's complete the table:

Final Answer: $A = 3$ and $B = 0$.

Task 7: Draw the graph of $y = -x + 1$

To draw the graph of $y = -x + 1$, we can use the table of values we just completed. The points are:

• $(-2, 3)$
• $(-1, 2)$
• $(0, 1)$
• $(1, 0)$
• $(2, -1)$

Plot these points on the coordinate plane and draw a straight line through them. The line should have a slope of $-1$ and a y-intercept of $1$.