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: $(0, 50)$.

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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: $(0, -6)$.

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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: $(0, -4)$.

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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: $(0, -15)$.

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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.