A line with gradient 2 intersects a line with gradient 6 at the point (40, 30). The distance between x-intercepts of these lines, is

1. Break Down the Problem

We need to find the x-intercepts of the two lines with gradients 2 and 6 that intersect at the point (40, 30). After finding the x-intercepts, we will calculate the distance between these two points.

2. Relevant Concepts

• The equation of a line can be written in the form $y = mx + b$, where $m$ is the gradient and $b$ is the y-intercept.
• The x-intercept of a line is found by setting $y = 0$ and solving for $x$.

3. Analysis and Detail

Step 3.1: Equation of the first line (gradient = 2)

Given that the gradient $m_1 = 2$ and it passes through (40, 30): $y - 30 = 2(x - 40)$ Simplifying: $y - 30 = 2x - 80 \implies y = 2x - 50$

Step 3.2: Find the x-intercept of the first line

Set $y = 0$: $0 = 2x - 50 \implies 2x = 50 \implies x = 25$ So, the x-intercept of the first line is $(25, 0)$.

Step 3.3: Equation of the second line (gradient = 6)

Given that the gradient $m_2 = 6$ and it also passes through (40, 30): $y - 30 = 6(x - 40)$ Simplifying: $y - 30 = 6x - 240 \implies y = 6x - 210$

Step 3.4: Find the x-intercept of the second line

Set $y = 0$: $0 = 6x - 210 \implies 6x = 210 \implies x = 35$ So, the x-intercept of the second line is $(35, 0)$.

4. Verify and Summarize

To find the distance between the x-intercepts $(25, 0)$ and $(35, 0)$: $\text{Distance} = |35 - 25| = 10$

Final Answer

The distance between the x-intercepts of the two lines is $10$ units.