Compute the area of the triangular region bounded by the y-axis and the lines. y=-x and y=-13x+9

To find the area of the triangular region bounded by the y-axis and the lines \( y = -x \) and \( y = -13x + 9 \), we can follow these steps:

### Step 1: Find the points of intersection
1. **Intersection with the y-axis**:
   - The y-axis is defined by \( x = 0 \).
   - For \( y = -x \), substituting \( x = 0 \) gives \( y = 0 \). So, the line \( y = -x \) intersects the y-axis at \( (0, 0) \).
   - For \( y = -13x + 9 \), substituting \( x = 0 \) gives \( y = 9 \). So, the line \( y = -13x + 9 \) intersects the y-axis at \( (0, 9) \).

2. **Intersection of the lines \( y = -x \) and \( y = -13x + 9 \)**:
   To find the intersection, set the equations equal to each other:
   \[
   -x = -13x + 9
   \]
   Solve for \( x \):
   \[
   -x + 13x = 9
   \]
   \[
   12x = 9 \quad \Rightarrow \quad x = \frac{9}{12} = \frac{3}{4}
   \]
   Now substitute \( x = \frac{3}{4} \) into \( y = -x \) to find \( y \):
   \[
   y = -\frac{3}{4}
   \]
   So, the two lines intersect at \( \left( \frac{3}{4}, -\frac{3}{4} \right) \).

### Step 2: Find the vertices of the triangle
The vertices of the triangle are:
- \( (0, 0) \): Intersection of the y-axis and \( y = -x \).
- \( (0, 9) \): Intersection of the y-axis and \( y = -13x + 9 \).
- \( \left( \frac{3}{4}, -\frac{3}{4} \right) \): Intersection of the two lines.

### Step 3: Use the formula for the area of a triangle
The area \( A \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\[
A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the vertices \( (0, 0), (0, 9), \left( \frac{3}{4}, -\frac{3}{4} \right) \):
\[
A = \frac{1}{2} \left| 0(9 - (-\frac{3}{4})) + 0(-\frac{3}{4} - 0) + \frac{3}{4}(0 - 9) \right|
\]
\[
A = \frac{1}{2} \left| \frac{3}{4} \times (-9) \right|
\]
\[
A = \frac{1}{2} \times \frac{27}{4} = \frac{27}{8}
\]
Thus, the area of the triangle is \( \frac{27}{8} \) square units, or 3.375 square units.

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Would you like further clarification on any step, or more details?

Here are 5 related questions:
1. How do you find the intersection point of two lines in general?
2. What is the significance of the absolute value in the area formula for a triangle?
3. How would the problem change if one of the lines was not linear but quadratic?
4. How can we use matrices to compute the area of a triangle from vertices?
5. What happens to the area if we shift the entire triangle vertically?

**Tip:** Always ensure that the vertices of your triangle are listed in a counterclockwise or clockwise direction to avoid errors in the area formula.

Find the area of the triangular region formed by the y-axis and the lines y=-x and y=-13x+9. This detailed solution walks through the intersection points, vertex calculations, and application of the triangle area formula. Suitable for students in Grades 9-12.

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