Show that the lines
`(x-1)/(2)=(y-2)/(3)=(z-3)/(4) " and " (x-4)/(5)=(y-1)/(2)=z`
intersect each other . Find their point of intersection.

Answer

To show that the lines given by the equations

1. \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}\) (Line 1)

2. \(\frac{x-4}{5} = \frac{y-1}{2} = z\) (Line 2)

intersect and to find their point of intersection, we can follow these steps:

### Step 1: Parameterize the lines
Let's denote the parameter for Line 1 as \(\lambda\) and for Line 2 as \(\mu\).

For Line 1:
- From \(\frac{x-1}{2} = \lambda\), we get \(x = 2\lambda + 1\)
- From \(\frac{y-2}{3} = \lambda\), we get \(y = 3\lambda + 2\)
- From \(\frac{z-3}{4} = \lambda\), we get \(z = 4\lambda + 3\)

Thus, a point on Line 1 can be represented as:
\[ (x, y, z) = (2\lambda + 1, 3\lambda + 2, 4\lambda + 3) \]

For Line 2:
- From \(\frac{x-4}{5} = \mu\), we get \(x = 5\mu + 4\)
- From \(\frac{y-1}{2} = \mu\), we get \(y = 2\mu + 1\)
- From \(z = \mu\), we have \(z = \mu\)

Thus, a point on Line 2 can be represented as:
\[ (x, y, z) = (5\mu + 4, 2\mu + 1, \mu) \]

### Step 2: Set the points equal
Since we want to find the intersection, we set the coordinates equal to each other:
\[ (2\lambda + 1, 3\lambda + 2, 4\lambda + 3) = (5\mu + 4, 2\mu + 1, \mu) \]

This gives us three equations:
1. \(2\lambda + 1 = 5\mu + 4\) (Equation 1)
2. \(3\lambda + 2 = 2\mu + 1\) (Equation 2)
3. \(4\lambda + 3 = \mu\) (Equation 3)

### Step 3: Solve the equations
We will solve these equations step by step.

**From Equation 3:**
\[ \mu = 4\lambda + 3 \]

**Substituting \(\mu\) in Equation 1:**
\[ 2\lambda + 1 = 5(4\lambda + 3) + 4 \]
\[ 2\lambda + 1 = 20\lambda + 15 + 4 \]
\[ 2\lambda + 1 = 20\lambda + 19 \]
\[ 2\lambda - 20\lambda = 19 - 1 \]
\[ -18\lambda = 18 \implies \lambda = -1 \]

**Substituting \(\lambda\) back to find \(\mu\):**
\[ \mu = 4(-1) + 3 = -4 + 3 = -1 \]

### Step 4: Find the point of intersection
Now that we have \(\lambda = -1\) and \(\mu = -1\), we can substitute these values back into either line's parameterization to find the point of intersection.

Using Line 1:
\[ x = 2(-1) + 1 = -2 + 1 = -1 \]
\[ y = 3(-1) + 2 = -3 + 2 = -1 \]
\[ z = 4(-1) + 3 = -4 + 3 = -1 \]

Thus, the point of intersection is:
\[ (-1, -1, -1) \]

### Conclusion
The lines intersect at the point \((-1, -1, -1)\).