Write down the intersections with the x and y-axes respectively for each of the following lines: a) x + 2y =100 b) 4x +5y = -30 c) 10x -5y = 20 d) x = y +15

To find the intersections with the x-axis and y-axis for each of the given lines, we follow these steps:

- **For the x-axis intersection**, set \( y = 0 \) and solve for \( x \).
- **For the y-axis intersection**, set \( x = 0 \) and solve for \( y \).

### a) \( x + 2y = 100 \)

1. **X-axis intersection** (Set \( y = 0 \)):
   \[
   x + 2(0) = 100 \implies x = 100
   \]
   Intersection with the x-axis: \( (100, 0) \).

2. **Y-axis intersection** (Set \( x = 0 \)):
   \[
   0 + 2y = 100 \implies y = \frac{100}{2} = 50
   \]
   Intersection with the y-axis: \( (0, 50) \).

---

### b) \( 4x + 5y = -30 \)

1. **X-axis intersection** (Set \( y = 0 \)):
   \[
   4x + 5(0) = -30 \implies 4x = -30 \implies x = \frac{-30}{4} = -7.5
   \]
   Intersection with the x-axis: \( (-7.5, 0) \).

2. **Y-axis intersection** (Set \( x = 0 \)):
   \[
   4(0) + 5y = -30 \implies 5y = -30 \implies y = \frac{-30}{5} = -6
   \]
   Intersection with the y-axis: \( (0, -6) \).

---

### c) \( 10x - 5y = 20 \)

1. **X-axis intersection** (Set \( y = 0 \)):
   \[
   10x - 5(0) = 20 \implies 10x = 20 \implies x = \frac{20}{10} = 2
   \]
   Intersection with the x-axis: \( (2, 0) \).

2. **Y-axis intersection** (Set \( x = 0 \)):
   \[
   10(0) - 5y = 20 \implies -5y = 20 \implies y = \frac{-20}{5} = -4
   \]
   Intersection with the y-axis: \( (0, -4) \).

---

### d) \( x = y + 15 \)

1. **X-axis intersection** (Set \( y = 0 \)):
   \[
   x = 0 + 15 \implies x = 15
   \]
   Intersection with the x-axis: \( (15, 0) \).

2. **Y-axis intersection** (Set \( x = 0 \)):
   \[
   0 = y + 15 \implies y = -15
   \]
   Intersection with the y-axis: \( (0, -15) \).

---

### Final answers:
- **a)** X-axis: \( (100, 0) \), Y-axis: \( (0, 50) \)
- **b)** X-axis: \( (-7.5, 0) \), Y-axis: \( (0, -6) \)
- **c)** X-axis: \( (2, 0) \), Y-axis: \( (0, -4) \)
- **d)** X-axis: \( (15, 0) \), Y-axis: \( (0, -15) \)

Let me know if you'd like more details or have any questions!  
Here are 5 follow-up questions:

1. How would you find the slope of the line \( x + 2y = 100 \)?
2. What does the intersection point represent geometrically?
3. How would you find the intersection of two lines, for example \( 4x + 5y = -30 \) and \( 10x - 5y = 20 \)?
4. How does the equation \( x = y + 15 \) differ from the others in terms of graphing?
5. What would be the general form of a line equation?

**Tip:** The x-intercept is found by setting \( y = 0 \) because any point on the x-axis has a \( y \)-coordinate of zero.

Learn how to find the x and y intercepts for linear equations such as x + 2y = 100, 4x + 5y = -30, and more. This guide provides a detailed, step-by-step approach suitable for students in Grades 8-10, covering essential algebraic concepts and intercept theorem applications.

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